Solving LinkedIn Patches by Linear Optimization

Problem Modeling¶

Ranges¶

Basically, we’ll have three ranges for defining all the next components: a range for rows, columns and rectangles.

$I = \{1, \cdots, m\}$

The row range, where $m$ is the total amount of rows on game grid.

$J = \{1, \cdots, n\}$

The column range, where $n$ is the total amount of columns on game grid.

$K = \{1, \cdots, p\}$

Range of all patches to be drawn on grid game, where $p$ is the total of patch seeds on grid.

Sets¶

From $I$, $J$ and $K$ ranges, we can define the following sets:

$S = I \times J = \{(i,j) \mid \forall i \in I, \forall j \in J\}$

Set of all grid squares, that is just the Cartesian product between sets $I$ and $J$.

$E = \{(i,j,k) \mid i \in I, j \in J, k \in K\} \subset I \times J \times K$

Set of all squares $(i,j)$ that carry a seed about a patch $k$.

$V \subseteq K$

Set of required vertical patches, i. e., patches whose height is greater than its width.

$H \subseteq K$

Set of required horizontal patches, i. e., patches whose width is greater than its height.

$Q \subseteq K$

Set of required square patches, i. e., patches whose width is equal to its height.

$A \subseteq K$

Set of patches with a required area $a_k$.

Decision Variables¶

For Patches game, it’ll have seven sets of decision varibles.

Integer Variables¶

The following integer variables will be used to define the position and dimensions of the patches, which are the set of variables that solves the game:

$l_k \in I, \quad \forall k \in K$

Index of the leftmost column of the patch $k$.

$t_k \in J, \quad \forall k \in K$

Index of the top row of the patch $k$.

$w_k \in I, \quad \forall k \in K$

Width of the patch $k$.

$h_k \in J, \quad \forall k \in K$

Height of the patch $k$.

Binary Variables¶

However, in order to ensure the rectangularity and continuity of the geometric shapes and their required areas, it is necessary to use the following auxiliary binary variables:

$u_{ik} \in \B, \quad \forall (i,k) \in I \times K$

$u_{ik} = 1$, if the row $i$ passes through the patch $k$

$u_{ik} = 0$, otherwise

$v_{jk} \in \B, \quad \forall (j,k) \in J \times K$

$v_{jk} = 1$, if the column $j$ passes through the patch $k$

$v_{jk} = 0$, otherwise

$x_{ijk} \in \B, \quad \forall (i,j,k) \in I \times J \times K$

$x_{ijk} = 1$, if the square $(i,j)$ will be covered by patch $k$.

$x_{ijk} = 0$, otherwise

Parameters¶

$a_k \in \N$

Required area for the patch $k \in A$.

Objective Function¶

Since we only want to find a solution that satisfies all the rules of the game, the objective function of Patches’ model is a constant. Therefore, the LOP is a feasibility problem, whose objective function consists of maximizing (or minimizing) an arbitrary constant.

Constraints¶

Domain Constraints

First, let’s settle the sets of constraints for the domain of decision variables.

Integrity Constraints

The main decision variables must be integers.

$$l_k \in \N, \quad \forall k \in K$$

(2)

$$t_k \in \N, \quad \forall k \in K$$

(3)

$$w_k \in \N, \quad \forall k \in K$$

(4)

$$h_k \in \N, \quad \forall k \in K$$

(5)

Binarity Constraints

The auxiliary decision variables must be binary.

$$u_{ik} \in \B, \quad \forall (i,k) \in I \times K$$

(6)

$$v_{jk} \in \B, \quad \forall (j,k) \in J \times K$$

(7)

$$x_{ijk} \in \B, \quad \forall (i,j,k) \in I \times J \times K$$

(8)

Unique-Patch-Per-Square Constraints

Each grid square $(i,j)$ must be covered by only one patch $k$.

$$\sum_{k \in K}{x_{ijk}} = 1, \quad \forall (i,j) \in S$$

(9)

Grid-Boundaries Constraints

These sets of constraints ensure that any patch $k$ lies within the grid’s row and column boundaries.

Top-Row-Position Constraints

For each patch $k$, the position $t_k$ of its top row must be equal or greater than 1.

$$t_k \ge 1, \quad \forall k \in K$$

(10)

Bottom-Row-Position Constraints

For each patch $k$, the position of its bottom row, given by the expression $t_k + h_k - 1$, must be equal or less than the total amount of rows $n$.

$$t_k + h_k - 1 \le n, \quad \forall k \in K$$

(11)

Leftmost-Column-Position Constraints

For each patch $k$, the position $l_k$ of its leftmost column must be equal or greater than 1.

$$l_k \ge 1, \quad \forall k \in K$$

(12)

Rightmost-Column-Position Constraints

For each patch $k$, the position of its rightmost column, given by the expression $l_k + w_k - 1$, must be equal or less than the total amount of columns $m$.

$$l_k + w_k - 1 \le m, \quad \forall k \in K$$

(13)

Patch-Boundaries Constraints

These sets of constraints uses the Big $M$ strategy to ensure that any row $i$ and column $j$ that passes through a patch $k$ lies within its boundaries.

Top-Boundary Constraints

if a row $i$ passes through a patch $k$, then the index $i$ must be equal or greater than its top row’s index $t_k$.

$$t_k - i \le M(1 - u_{ik}), \quad \forall (i,k) \in I \times K$$

(14)

Bottom-Boundary Constraints

if a square $(i,j)$ is covered by patch $k$, then the index $i$ must be equal or less than the patch’s bottom row’s index, given by $t_k + h_k - 1$.

$$i - (t_k + h_k - 1) \le M(1 - u_{ik}), \quad \forall (i,k) \in I \times K$$

(15)

Leftmost-Boundary Constraints

if a colum $j$ passes through patch $k$, then the index $j$ must be equal or greater than patch’s rightmost column’s index $l_k$.

$$l_k - j \le M(1 - v_{jk}), \quad \forall (j,k) \in J \times K$$

(16)

Rightmost-Boundary Constraints

if a colum $j$ passes through patch $k$, then the index $j$ must be equal or less the patch’s last column’s index, given by $l_k + w_k - 1$.

$$j - (l_k + w_k - 1) \le M(1 - v_{jk}), \quad \forall (j,k) \in J \times K$$

(17)

Patch Dimensions constraints

These constraints are important to ensure that there’ll not be any gap inside the patches.

Height Constraints

The sum of all selected rows $u_{ik}$ for a patch $k$ must be equal to its height.

$$\sum_{i \in I}{u_{ik}} = h_k, \quad \forall k \in K$$

(18)

Width Constraints

The sum of all selected columns $v_{jk}$ for a patch $k$ must be equal to its width.

$$\sum_{j \in J}{v_{jk}} = w_k, \quad \forall k \in K$$

(19)

McCormick Linearization Constraints

These sets of constraints translates the non-linear contraint $x_{ijk} = u_{ik} \times v_{jk}$ between the binary varibles in the model into three sets of linear constraints. They will ensure that if a row $i$ and a column $j$ pass through a patch $k$, then the square $(i,j)$ will must be covered by that patch $k$. Also, if $u_{ik}=0$ or $v_{jk}=0$, then $x_{ijk}$ must be zero, which will ensure that if a row or column does not pass through a patch, then the square $(i,j)$ will not be covered by it.

Cutout-Row Constraints

If a row $i$ doesn’t pass through a patch $k$, then all squares in that row must not be covered by that patch.

$$\sum_{j \in J}{x_{ijk}} \le n \cdot u_{ik}, \quad \forall (i,k) \in I \times K$$

(20)

Cutout-Column Constraints

If a column $j$ doesn’t pass through a patch $k$, then all squares in that column must not be covered by that patch.

$$\sum_{i \in I}{x_{ijk}} \le m \cdot v_{jk}, \quad \forall (j,k) \in J \times K$$

(21)

Square-Activator Constraints

If both row $i$ and column $j$ pass the patch $k$, then the square at this intersection must be covered by that patch.

$$x_{ijk} \ge u_{ik} + v_{jk} - 1, \quad \forall (i,j,k) \in I \times J \times K$$

(22)

Seed Square Constraints

These sets of constraints deal with prescriptions required by the seed squares.

Seed Square Coverage Constraints

For each square $(i,j)$ with a seed about patch $k$, it must be covered by $k$.

$$x_{ijk} = 1, \quad \forall (i,j,k) \in T$$

(23)

Area Constraints

For each patch $k$ with a predefined area $a_k$, the sum of its covered squares $x_{ijk}$ must be equal to $a_k$

$$\sum_{(i,j) \in S}{x_{ijk}} = a_k, \quad \forall k \in A$$

(24)

Vertical Patch Constraints

For each vertical patch $k$, its height $h_k$ must be greater than its width $w_k$.

$$w_k < h_k, \quad \forall k \in V$$

(25)

Horizontal Patch Constraints

For each horizontal patch $k$, its width $w_k$ must be greater than its height $h_k$.

$$w_k > h_k, \quad \forall k \in H$$

(26)

Square Patch Constraints

For each square patch $k$, its width $w_k$ must be equal to its height $h_k$.

$$w_k = h_k, \quad \forall k \in Q$$

(27)

Abstract Model¶

With all components defined, we have the following abstract model for Patches game:

Concrete model¶

The Patches game to be solved will be the No. 16, published on April 2^nd, 2026.

And you concrete model is as the following:

S.t.:

Unique-Patch-Per-Square Constraints

$x_{11\yellowbox} + x_{11\tealbox} + x_{11\purplebox} + x_{11\greenbox} + x_{11\orangebox} + x_{11\redbox} + x_{11\bluebox} + x_{11\magentabox} + x_{11\brickbox} + x_{11\brownbox} = 1$ _{Square (1,1)}

$x_{12\yellowbox} + x_{12\tealbox} + x_{12\purplebox} + x_{12\greenbox} + x_{12\orangebox} + x_{12\redbox} + x_{12\bluebox} + x_{12\magentabox} + x_{12\brickbox} + x_{12\brownbox} = 1$ _{Square (1,2)}

$x_{13\yellowbox} + x_{13\tealbox} + x_{13\purplebox} + x_{13\greenbox} + x_{13\orangebox} + x_{13\redbox} + x_{13\bluebox} + x_{13\magentabox} + x_{13\brickbox} + x_{13\brownbox} = 1$ _{Square (1,3)}

$x_{14\yellowbox} + x_{14\tealbox} + x_{14\purplebox} + x_{14\greenbox} + x_{14\orangebox} + x_{14\redbox} + x_{14\bluebox} + x_{14\magentabox} + x_{14\brickbox} + x_{14\brownbox} = 1$ _{Square (1,4)}

$x_{15\yellowbox} + x_{15\tealbox} + x_{15\purplebox} + x_{15\greenbox} + x_{15\orangebox} + x_{15\redbox} + x_{15\bluebox} + x_{15\magentabox} + x_{15\brickbox} + x_{15\brownbox} = 1$ _{Square (1,5)}

$x_{16\yellowbox} + x_{16\tealbox} + x_{16\purplebox} + x_{16\greenbox} + x_{16\orangebox} + x_{16\redbox} + x_{16\bluebox} + x_{16\magentabox} + x_{16\brickbox} + x_{16\brownbox} = 1$ _{Square (1,6)}

$x_{21\yellowbox} + x_{21\tealbox} + x_{21\purplebox} + x_{21\greenbox} + x_{21\orangebox} + x_{21\redbox} + x_{21\bluebox} + x_{21\magentabox} + x_{21\brickbox} + x_{21\brownbox} = 1$ _{Square (2,1)}

$x_{22\yellowbox} + x_{22\tealbox} + x_{22\purplebox} + x_{22\greenbox} + x_{22\orangebox} + x_{22\redbox} + x_{22\bluebox} + x_{22\magentabox} + x_{22\brickbox} + x_{22\brownbox} = 1$ _{Square (2,2)}

$x_{23\yellowbox} + x_{23\tealbox} + x_{23\purplebox} + x_{23\greenbox} + x_{23\orangebox} + x_{23\redbox} + x_{23\bluebox} + x_{23\magentabox} + x_{23\brickbox} + x_{23\brownbox} = 1$ _{Square (2,3)}

$x_{24\yellowbox} + x_{24\tealbox} + x_{24\purplebox} + x_{24\greenbox} + x_{24\orangebox} + x_{24\redbox} + x_{24\bluebox} + x_{24\magentabox} + x_{24\brickbox} + x_{24\brownbox} = 1$ _{Square (2,4)}

$x_{25\yellowbox} + x_{25\tealbox} + x_{25\purplebox} + x_{25\greenbox} + x_{25\orangebox} + x_{25\redbox} + x_{25\bluebox} + x_{25\magentabox} + x_{25\brickbox} + x_{25\brownbox} = 1$ _{Square (2,5)}

$x_{26\yellowbox} + x_{26\tealbox} + x_{26\purplebox} + x_{26\greenbox} + x_{26\orangebox} + x_{26\redbox} + x_{26\bluebox} + x_{26\magentabox} + x_{26\brickbox} + x_{26\brownbox} = 1$ _{Square (2,6)}

$x_{31\yellowbox} + x_{31\tealbox} + x_{31\purplebox} + x_{31\greenbox} + x_{31\orangebox} + x_{31\redbox} + x_{31\bluebox} + x_{31\magentabox} + x_{31\brickbox} + x_{31\brownbox} = 1$ _{Square (3,1)}

$x_{32\yellowbox} + x_{32\tealbox} + x_{32\purplebox} + x_{32\greenbox} + x_{32\orangebox} + x_{32\redbox} + x_{32\bluebox} + x_{32\magentabox} + x_{32\brickbox} + x_{32\brownbox} = 1$ _{Square (3,2)}

$x_{33\yellowbox} + x_{33\tealbox} + x_{33\purplebox} + x_{33\greenbox} + x_{33\orangebox} + x_{33\redbox} + x_{33\bluebox} + x_{33\magentabox} + x_{33\brickbox} + x_{33\brownbox} = 1$ _{Square (3,3)}

$x_{34\yellowbox} + x_{34\tealbox} + x_{34\purplebox} + x_{34\greenbox} + x_{34\orangebox} + x_{34\redbox} + x_{34\bluebox} + x_{34\magentabox} + x_{34\brickbox} + x_{34\brownbox} = 1$ _{Square (3,4)}

$x_{35\yellowbox} + x_{35\tealbox} + x_{35\purplebox} + x_{35\greenbox} + x_{35\orangebox} + x_{35\redbox} + x_{35\bluebox} + x_{35\magentabox} + x_{35\brickbox} + x_{35\brownbox} = 1$ _{Square (3,5)}

$x_{36\yellowbox} + x_{36\tealbox} + x_{36\purplebox} + x_{36\greenbox} + x_{36\orangebox} + x_{36\redbox} + x_{36\bluebox} + x_{36\magentabox} + x_{36\brickbox} + x_{36\brownbox} = 1$ _{Square (3,6)}

$x_{41\yellowbox} + x_{41\tealbox} + x_{41\purplebox} + x_{41\greenbox} + x_{41\orangebox} + x_{41\redbox} + x_{41\bluebox} + x_{41\magentabox} + x_{41\brickbox} + x_{41\brownbox} = 1$ _{Square (4,1)}

$x_{42\yellowbox} + x_{42\tealbox} + x_{42\purplebox} + x_{42\greenbox} + x_{42\orangebox} + x_{42\redbox} + x_{42\bluebox} + x_{42\magentabox} + x_{42\brickbox} + x_{42\brownbox} = 1$ _{Square (4,2)}

$x_{43\yellowbox} + x_{43\tealbox} + x_{43\purplebox} + x_{43\greenbox} + x_{43\orangebox} + x_{43\redbox} + x_{43\bluebox} + x_{43\magentabox} + x_{43\brickbox} + x_{43\brownbox} = 1$ _{Square (4,3)}

$x_{44\yellowbox} + x_{44\tealbox} + x_{44\purplebox} + x_{44\greenbox} + x_{44\orangebox} + x_{44\redbox} + x_{44\bluebox} + x_{44\magentabox} + x_{44\brickbox} + x_{44\brownbox} = 1$ _{Square (4,4)}

$x_{45\yellowbox} + x_{45\tealbox} + x_{45\purplebox} + x_{45\greenbox} + x_{45\orangebox} + x_{45\redbox} + x_{45\bluebox} + x_{45\magentabox} + x_{45\brickbox} + x_{45\brownbox} = 1$ _{Square (4,5)}

$x_{46\yellowbox} + x_{46\tealbox} + x_{46\purplebox} + x_{46\greenbox} + x_{46\orangebox} + x_{46\redbox} + x_{46\bluebox} + x_{46\magentabox} + x_{46\brickbox} + x_{46\brownbox} = 1$ _{Square (4,6)}

$x_{51\yellowbox} + x_{51\tealbox} + x_{51\purplebox} + x_{51\greenbox} + x_{51\orangebox} + x_{51\redbox} + x_{51\bluebox} + x_{51\magentabox} + x_{51\brickbox} + x_{51\brownbox} = 1$ _{Square (5,1)}

$x_{52\yellowbox} + x_{52\tealbox} + x_{52\purplebox} + x_{52\greenbox} + x_{52\orangebox} + x_{52\redbox} + x_{52\bluebox} + x_{52\magentabox} + x_{52\brickbox} + x_{52\brownbox} = 1$ _{Square (5,2)}

$x_{53\yellowbox} + x_{53\tealbox} + x_{53\purplebox} + x_{53\greenbox} + x_{53\orangebox} + x_{53\redbox} + x_{53\bluebox} + x_{53\magentabox} + x_{53\brickbox} + x_{53\brownbox} = 1$ _{Square (5,3)}

$x_{54\yellowbox} + x_{54\tealbox} + x_{54\purplebox} + x_{54\greenbox} + x_{54\orangebox} + x_{54\redbox} + x_{54\bluebox} + x_{54\magentabox} + x_{54\brickbox} + x_{54\brownbox} = 1$ _{Square (5,4)}

$x_{55\yellowbox} + x_{55\tealbox} + x_{55\purplebox} + x_{55\greenbox} + x_{55\orangebox} + x_{55\redbox} + x_{55\bluebox} + x_{55\magentabox} + x_{55\brickbox} + x_{55\brownbox} = 1$ _{Square (5,5)}

$x_{56\yellowbox} + x_{56\tealbox} + x_{56\purplebox} + x_{56\greenbox} + x_{56\orangebox} + x_{56\redbox} + x_{56\bluebox} + x_{56\magentabox} + x_{56\brickbox} + x_{56\brownbox} = 1$ _{Square (5,6)}

$x_{61\yellowbox} + x_{61\tealbox} + x_{61\purplebox} + x_{61\greenbox} + x_{61\orangebox} + x_{61\redbox} + x_{61\bluebox} + x_{61\magentabox} + x_{61\brickbox} + x_{61\brownbox} = 1$ _{Square (6,1)}

$x_{62\yellowbox} + x_{62\tealbox} + x_{62\purplebox} + x_{62\greenbox} + x_{62\orangebox} + x_{62\redbox} + x_{62\bluebox} + x_{62\magentabox} + x_{62\brickbox} + x_{62\brownbox} = 1$ _{Square (6,2)}

$x_{63\yellowbox} + x_{63\tealbox} + x_{63\purplebox} + x_{63\greenbox} + x_{63\orangebox} + x_{63\redbox} + x_{63\bluebox} + x_{63\magentabox} + x_{63\brickbox} + x_{63\brownbox} = 1$ _{Square (6,3)}

$x_{64\yellowbox} + x_{64\tealbox} + x_{64\purplebox} + x_{64\greenbox} + x_{64\orangebox} + x_{64\redbox} + x_{64\bluebox} + x_{64\magentabox} + x_{64\brickbox} + x_{64\brownbox} = 1$ _{Square (6,4)}

$x_{65\yellowbox} + x_{65\tealbox} + x_{65\purplebox} + x_{65\greenbox} + x_{65\orangebox} + x_{65\redbox} + x_{65\bluebox} + x_{65\magentabox} + x_{65\brickbox} + x_{65\brownbox} = 1$ _{Square (6,5)}

$x_{66\yellowbox} + x_{66\tealbox} + x_{66\purplebox} + x_{66\greenbox} + x_{66\orangebox} + x_{66\redbox} + x_{66\bluebox} + x_{66\magentabox} + x_{66\brickbox} + x_{66\brownbox} = 1$ _{Square (6,6)}

Top Row Constraints

$t_{\yellowbox} \ge 1$ _{Yellow Patch}

$t_{\tealbox} \ge 1$ _{Teal Patch}

$t_{\purplebox} \ge 1$ _{Purple Patch}

$t_{\greenbox} \ge 1$ _{Green Patch}

$t_{\orangebox} \ge 1$ _{Orange Patch}

$t_{\redbox} \ge 1$ _{Red Patch}

$t_{\bluebox} \ge 1$ _{Blue Patch}

$t_{\magentabox} \ge 1$ _{Magenta Patch}

$t_{\brickbox} \ge 1$ _{Brick Patch}

$t_{\brownbox} \ge 1$ _{Brown Patch}

Bottom Row Constraints

$t_{\yellowbox} + h_{\yellowbox} - 1 \le 6$ _{Yellow Patch}

$t_{\tealbox} + h_{\tealbox} - 1 \le 6$ _{Teal Patch}

$t_{\purplebox} + h_{\purplebox} - 1 \le 6$ _{Purple Patch}

$t_{\greenbox} + h_{\greenbox} - 1 \le 6$ _{Green Patch}

$t_{\orangebox} + h_{\orangebox} - 1 \le 6$ _{Orange Patch}

$t_{\redbox} + h_{\redbox} - 1 \le 6$ _{Red Patch}

$t_{\bluebox} + h_{\bluebox} - 1 \le 6$ _{Blue Patch}

$t_{\magentabox} + h_{\magentabox} - 1 \le 6$ _{Magenta Patch}

$t_{\brickbox} + h_{\brickbox} - 1 \le 6$ _{Brick Patch}

$t_{\brownbox} + h_{\brownbox} - 1 \le 6$ _{Brown Patch}

Leftmost Column Constraints

$l_{\yellowbox} \ge 1$ _{Yellow Patch}

$l_{\tealbox} \ge 1$ _{Teal Patch}

$l_{\purplebox} \ge 1$ _{Purple Patch}

$l_{\greenbox} \ge 1$ _{Green Patch}

$l_{\orangebox} \ge 1$ _{Orange Patch}

$l_{\redbox} \ge 1$ _{Red Patch}

$l_{\bluebox} \ge 1$ _{Blue Patch}

$l_{\magentabox} \ge 1$ _{Magenta Patch}

$l_{\brickbox} \ge 1$ _{Brick Patch}

$l_{\brownbox} \ge 1$ _{Brown Patch}

Rightmost Column Constraints

$l_{\yellowbox} + w_{\yellowbox} - 1 \le 6$ _{Yellow Patch}

$l_{\tealbox} + w_{\tealbox} - 1 \le 6$ _{Teal Patch}

$l_{\purplebox} + w_{\purplebox} - 1 \le 6$ _{Purple Patch}

$l_{\greenbox} + w_{\greenbox} - 1 \le 6$ _{Green Patch}

$l_{\orangebox} + w_{\orangebox} - 1 \le 6$ _{Orange Patch}

$l_{\redbox} + w_{\redbox} - 1 \le 6$ _{Red Patch}

$l_{\bluebox} + w_{\bluebox} - 1 \le 6$ _{Blue Patch}

$l_{\magentabox} + w_{\magentabox} - 1 \le 6$ _{Magenta Patch}

$l_{\brickbox} + w_{\brickbox} - 1 \le 6$ _{Brick Patch}

$l_{\brownbox} + w_{\brownbox} - 1 \le 6$ _{Brown Patch}

Top Boundary Constraints

$t_{\yellowbox} - 1 \le 6(1-u_{1\yellowbox})$ _{Row 1, Yellow Patch}

$t_{\yellowbox} - 2 \le 6(1-u_{2\yellowbox})$ _{Row 2, Yellow Patch}

$t_{\yellowbox} - 3 \le 6(1-u_{3\yellowbox})$ _{Row 3, Yellow Patch}

$t_{\yellowbox} - 4 \le 6(1-u_{4\yellowbox})$ _{Row 4, Yellow Patch}

$t_{\yellowbox} - 5 \le 6(1-u_{5\yellowbox})$ _{Row 5, Yellow Patch}

$t_{\yellowbox} - 6 \le 6(1-u_{6\yellowbox})$ _{Row 6, Yellow Patch}

$t_{\tealbox} - 1 \le 6(1-u_{1\tealbox})$ _{Row 1, Tealbox Patch}

$t_{\tealbox} - 2 \le 6(1-u_{2\tealbox})$ _{Row 2, Tealbox Patch}

$t_{\tealbox} - 3 \le 6(1-u_{3\tealbox})$ _{Row 3, Tealbox Patch}

$t_{\tealbox} - 4 \le 6(1-u_{4\tealbox})$ _{Row 4, Tealbox Patch}

$t_{\tealbox} - 5 \le 6(1-u_{5\tealbox})$ _{Row 5, Tealbox Patch}

$t_{\tealbox} - 6 \le 6(1-u_{6\tealbox})$ _{Row 6, Tealbox Patch}

$t_{\purplebox} - 1 \le 6(1-u_{1\purplebox})$ _{Row 1, Purple Patch}

$t_{\purplebox} - 2 \le 6(1-u_{2\purplebox})$ _{Row 2, Purple Patch}

$t_{\purplebox} - 3 \le 6(1-u_{3\purplebox})$ _{Row 3, Purple Patch}

$t_{\purplebox} - 4 \le 6(1-u_{4\purplebox})$ _{Row 4, Purple Patch}

$t_{\purplebox} - 5 \le 6(1-u_{5\purplebox})$ _{Row 5, Purple Patch}

$t_{\purplebox} - 6 \le 6(1-u_{6\purplebox})$ _{Row 6, Purple Patch}

$t_{\greenbox} - 1 \le 6(1-u_{1\greenbox})$ _{Row 1, Green Patch}

$t_{\greenbox} - 2 \le 6(1-u_{2\greenbox})$ _{Row 2, Green Patch}

$t_{\greenbox} - 3 \le 6(1-u_{3\greenbox})$ _{Row 3, Green Patch}

$t_{\greenbox} - 4 \le 6(1-u_{4\greenbox})$ _{Row 4, Green Patch}

$t_{\greenbox} - 5 \le 6(1-u_{5\greenbox})$ _{Row 5, Green Patch}

$t_{\greenbox} - 6 \le 6(1-u_{6\greenbox})$ _{Row 6, Green Patch}

$t_{\orangebox} - 1 \le 6(1-u_{1\orangebox})$ _{Row 1, Orange Patch}

$t_{\orangebox} - 2 \le 6(1-u_{2\orangebox})$ _{Row 2, Orange Patch}

$t_{\orangebox} - 3 \le 6(1-u_{3\orangebox})$ _{Row 3, Orange Patch}

$t_{\orangebox} - 4 \le 6(1-u_{4\orangebox})$ _{Row 4, Orange Patch}

$t_{\orangebox} - 5 \le 6(1-u_{5\orangebox})$ _{Row 5, Orange Patch}

$t_{\orangebox} - 6 \le 6(1-u_{6\orangebox})$ _{Row 6, Orange Patch}

$t_{\redbox} - 1 \le 6(1-u_{1\redbox})$ _{Row 1, Red Patch}

$t_{\redbox} - 2 \le 6(1-u_{2\redbox})$ _{Row 2, Red Patch}

$t_{\redbox} - 3 \le 6(1-u_{3\redbox})$ _{Row 3, Red Patch}

$t_{\redbox} - 4 \le 6(1-u_{4\redbox})$ _{Row 4, Red Patch}

$t_{\redbox} - 5 \le 6(1-u_{5\redbox})$ _{Row 5, Red Patch}

$t_{\redbox} - 6 \le 6(1-u_{6\redbox})$ _{Row 6, Red Patch}

$t_{\bluebox} - 1 \le 6(1-u_{1\bluebox})$ _{Row 1, Blue Patch}

$t_{\bluebox} - 2 \le 6(1-u_{2\bluebox})$ _{Row 2, Blue Patch}

$t_{\bluebox} - 3 \le 6(1-u_{3\bluebox})$ _{Row 3, Blue Patch}

$t_{\bluebox} - 4 \le 6(1-u_{4\bluebox})$ _{Row 4, Blue Patch}

$t_{\bluebox} - 5 \le 6(1-u_{5\bluebox})$ _{Row 5, Blue Patch}

$t_{\bluebox} - 6 \le 6(1-u_{6\bluebox})$ _{Row 6, Blue Patch}

$t_{\magentabox} - 1 \le 6(1-u_{1\magentabox})$ _{Row 1, Magenta Patch}

$t_{\magentabox} - 2 \le 6(1-u_{2\magentabox})$ _{Row 2, Magenta Patch}

$t_{\magentabox} - 3 \le 6(1-u_{3\magentabox})$ _{Row 3, Magenta Patch}

$t_{\magentabox} - 4 \le 6(1-u_{4\magentabox})$ _{Row 4, Magenta Patch}

$t_{\magentabox} - 5 \le 6(1-u_{5\magentabox})$ _{Row 5, Magenta Patch}

$t_{\magentabox} - 6 \le 6(1-u_{6\magentabox})$ _{Row 6, Magenta Patch}

$t_{\brickbox} - 1 \le 6(1-u_{1\brickbox})$ _{Row 1, Brick Patch}

$t_{\brickbox} - 2 \le 6(1-u_{2\brickbox})$ _{Row 2, Brick Patch}

$t_{\brickbox} - 3 \le 6(1-u_{3\brickbox})$ _{Row 3, Brick Patch}

$t_{\brickbox} - 4 \le 6(1-u_{4\brickbox})$ _{Row 4, Brick Patch}

$t_{\brickbox} - 5 \le 6(1-u_{5\brickbox})$ _{Row 5, Brick Patch}

$t_{\brickbox} - 6 \le 6(1-u_{6\brickbox})$ _{Row 6, Brick Patch}

$t_{\brownbox} - 1 \le 6(1-u_{1\brownbox})$ _{Row 1, Brown Patch}

$t_{\brownbox} - 2 \le 6(1-u_{2\brownbox})$ _{Row 2, Brown Patch}

$t_{\brownbox} - 3 \le 6(1-u_{3\brownbox})$ _{Row 3, Brown Patch}

$t_{\brownbox} - 4 \le 6(1-u_{4\brownbox})$ _{Row 4, Brown Patch}

$t_{\brownbox} - 5 \le 6(1-u_{5\brownbox})$ _{Row 5, Brown Patch}

$t_{\brownbox} - 6 \le 6(1-u_{6\brownbox})$ _{Row 6, Brown Patch}

Bottom Boundary Constraints

$1 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{1\yellowbox})$ _{Row 1, Yellow Patch}

$2 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{2\yellowbox})$ _{Row 2, Yellow Patch}

$3 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{3\yellowbox})$ _{Row 3, Yellow Patch}

$4 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{4\yellowbox})$ _{Row 4, Yellow Patch}

$5 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{5\yellowbox})$ _{Row 5, Yellow Patch}

$6 - (t_{\yellowbox} + h_{\yellowbox} - 1) \le 6(1 - u_{6\yellowbox})$ _{Row 6, Yellow Patch}

$1 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{1\tealbox})$ _{Row 1, Tealbox Patch}

$2 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{2\tealbox})$ _{Row 2, Tealbox Patch}

$3 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{3\tealbox})$ _{Row 3, Tealbox Patch}

$4 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{4\tealbox})$ _{Row 4, Tealbox Patch}

$5 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{5\tealbox})$ _{Row 5, Tealbox Patch}

$6 - (t_{\tealbox} + h_{\tealbox} - 1) \le 6(1 - u_{6\tealbox})$ _{Row 6, Tealbox Patch}

$1 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{1\purplebox})$ _{Row 1, Purple Patch}

$2 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{2\purplebox})$ _{Row 2, Purple Patch}

$3 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{3\purplebox})$ _{Row 3, Purple Patch}

$4 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{4\purplebox})$ _{Row 4, Purple Patch}

$5 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{5\purplebox})$ _{Row 5, Purple Patch}

$6 - (t_{\purplebox} + h_{\purplebox} - 1) \le 6(1 - u_{6\purplebox})$ _{Row 6, Purple Patch}

$1 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{1\greenbox})$ _{Row 1, Green Patch}

$2 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{2\greenbox})$ _{Row 2, Green Patch}

$3 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{3\greenbox})$ _{Row 3, Green Patch}

$4 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{4\greenbox})$ _{Row 4, Green Patch}

$5 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{5\greenbox})$ _{Row 5, Green Patch}

$6 - (t_{\greenbox} + h_{\greenbox} - 1) \le 6(1 - u_{6\greenbox})$ _{Row 6, Green Patch}

$1 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{1\orangebox})$ _{Row 1, Orange Patch}

$2 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{2\orangebox})$ _{Row 2, Orange Patch}

$3 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{3\orangebox})$ _{Row 3, Orange Patch}

$4 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{4\orangebox})$ _{Row 4, Orange Patch}

$5 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{5\orangebox})$ _{Row 5, Orange Patch}

$6 - (t_{\orangebox} + h_{\orangebox} - 1) \le 6(1 - u_{6\orangebox})$ _{Row 6, Orange Patch}

$1 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{1\redbox})$ _{Row 1, Red Patch}

$2 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{2\redbox})$ _{Row 2, Red Patch}

$3 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{3\redbox})$ _{Row 3, Red Patch}

$4 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{4\redbox})$ _{Row 4, Red Patch}

$5 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{5\redbox})$ _{Row 5, Red Patch}

$6 - (t_{\redbox} + h_{\redbox} - 1) \le 6(1 - u_{6\redbox})$ _{Row 6, Red Patch}

$1 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{1\bluebox})$ _{Row 1, Blue Patch}

$2 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{2\bluebox})$ _{Row 2, Blue Patch}

$3 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{3\bluebox})$ _{Row 3, Blue Patch}

$4 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{4\bluebox})$ _{Row 4, Blue Patch}

$5 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{5\bluebox})$ _{Row 5, Blue Patch}

$6 - (t_{\bluebox} + h_{\bluebox} - 1) \le 6(1 - u_{6\bluebox})$ _{Row 6, Blue Patch}

$1 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{1\magentabox})$ _{Row 1, Magenta Patch}

$2 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{2\magentabox})$ _{Row 2, Magenta Patch}

$3 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{3\magentabox})$ _{Row 3, Magenta Patch}

$4 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{4\magentabox})$ _{Row 4, Magenta Patch}

$5 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{5\magentabox})$ _{Row 5, Magenta Patch}

$6 - (t_{\magentabox} + h_{\magentabox} - 1) \le 6(1 - u_{6\magentabox})$ _{Row 6, Magenta Patch}

$1 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{1\brickbox})$ _{Row 1, Brick Patch}

$2 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{2\brickbox})$ _{Row 2, Brick Patch}

$3 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{3\brickbox})$ _{Row 3, Brick Patch}

$4 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{4\brickbox})$ _{Row 4, Brick Patch}

$5 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{5\brickbox})$ _{Row 5, Brick Patch}

$6 - (t_{\brickbox} + h_{\brickbox} - 1) \le 6(1 - u_{6\brickbox})$ _{Row 6, Brick Patch}

$1 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{1\brownbox})$ _{Row 1, Brown Patch}

$2 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{2\brownbox})$ _{Row 2, Brown Patch}

$3 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{3\brownbox})$ _{Row 3, Brown Patch}

$4 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{4\brownbox})$ _{Row 4, Brown Patch}

$5 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{5\brownbox})$ _{Row 5, Brown Patch}

$6 - (t_{\brownbox} + h_{\brownbox} - 1) \le 6(1 - u_{6\brownbox})$ _{Row 6, Brown Patch}

Left Boundary Constraints

$l_{\yellowbox} - 1 \le 6(1 - v_{1\yellowbox})$ _{Column 1, Yellow Patch}

$l_{\yellowbox} - 2 \le 6(1 - v_{2\yellowbox})$ _{Column 2, Yellow Patch}

$l_{\yellowbox} - 3 \le 6(1 - v_{3\yellowbox})$ _{Column 3, Yellow Patch}

$l_{\yellowbox} - 4 \le 6(1 - v_{4\yellowbox})$ _{Column 4, Yellow Patch}

$l_{\yellowbox} - 5 \le 6(1 - v_{5\yellowbox})$ _{Column 5, Yellow Patch}

$l_{\yellowbox} - 6 \le 6(1 - v_{6\yellowbox})$ _{Column 6, Yellow Patch}

$l_{\tealbox} - 1 \le 6(1 - v_{1\tealbox})$ _{Column 1, Tealbox Patch}

$l_{\tealbox} - 2 \le 6(1 - v_{2\tealbox})$ _{Column 2, Tealbox Patch}

$l_{\tealbox} - 3 \le 6(1 - v_{3\tealbox})$ _{Column 3, Tealbox Patch}

$l_{\tealbox} - 4 \le 6(1 - v_{4\tealbox})$ _{Column 4, Tealbox Patch}

$l_{\tealbox} - 5 \le 6(1 - v_{5\tealbox})$ _{Column 5, Tealbox Patch}

$l_{\tealbox} - 6 \le 6(1 - v_{6\tealbox})$ _{Column 6, Tealbox Patch}

$l_{\purplebox} - 1 \le 6(1 - v_{1\purplebox})$ _{Column 1, Purple Patch}

$l_{\purplebox} - 2 \le 6(1 - v_{2\purplebox})$ _{Column 2, Purple Patch}

$l_{\purplebox} - 3 \le 6(1 - v_{3\purplebox})$ _{Column 3, Purple Patch}

$l_{\purplebox} - 4 \le 6(1 - v_{4\purplebox})$ _{Column 4, Purple Patch}

$l_{\purplebox} - 5 \le 6(1 - v_{5\purplebox})$ _{Column 5, Purple Patch}

$l_{\purplebox} - 6 \le 6(1 - v_{6\purplebox})$ _{Column 6, Purple Patch}

$l_{\greenbox} - 1 \le 6(1 - v_{1\greenbox})$ _{Column 1, Green Patch}

$l_{\greenbox} - 2 \le 6(1 - v_{2\greenbox})$ _{Column 2, Green Patch}

$l_{\greenbox} - 3 \le 6(1 - v_{3\greenbox})$ _{Column 3, Green Patch}

$l_{\greenbox} - 4 \le 6(1 - v_{4\greenbox})$ _{Column 4, Green Patch}

$l_{\greenbox} - 5 \le 6(1 - v_{5\greenbox})$ _{Column 5, Green Patch}

$l_{\greenbox} - 6 \le 6(1 - v_{6\greenbox})$ _{Column 6, Green Patch}

$l_{\orangebox} - 1 \le 6(1 - v_{1\orangebox})$ _{Column 1, Orange Patch}

$l_{\orangebox} - 2 \le 6(1 - v_{2\orangebox})$ _{Column 2, Orange Patch}

$l_{\orangebox} - 3 \le 6(1 - v_{3\orangebox})$ _{Column 3, Orange Patch}

$l_{\orangebox} - 4 \le 6(1 - v_{4\orangebox})$ _{Column 4, Orange Patch}

$l_{\orangebox} - 5 \le 6(1 - v_{5\orangebox})$ _{Column 5, Orange Patch}

$l_{\orangebox} - 6 \le 6(1 - v_{6\orangebox})$ _{Column 6, Orange Patch}

$l_{\redbox} - 1 \le 6(1 - v_{1\redbox})$ _{Column 1, Red Patch}

$l_{\redbox} - 2 \le 6(1 - v_{2\redbox})$ _{Column 2, Red Patch}

$l_{\redbox} - 3 \le 6(1 - v_{3\redbox})$ _{Column 3, Red Patch}

$l_{\redbox} - 4 \le 6(1 - v_{4\redbox})$ _{Column 4, Red Patch}

$l_{\redbox} - 5 \le 6(1 - v_{5\redbox})$ _{Column 5, Red Patch}

$l_{\redbox} - 6 \le 6(1 - v_{6\redbox})$ _{Column 6, Red Patch}

$l_{\bluebox} - 1 \le 6(1 - v_{1\bluebox})$ _{Column 1, Blue Patch}

$l_{\bluebox} - 2 \le 6(1 - v_{2\bluebox})$ _{Column 2, Blue Patch}

$l_{\bluebox} - 3 \le 6(1 - v_{3\bluebox})$ _{Column 3, Blue Patch}

$l_{\bluebox} - 4 \le 6(1 - v_{4\bluebox})$ _{Column 4, Blue Patch}

$l_{\bluebox} - 5 \le 6(1 - v_{5\bluebox})$ _{Column 5, Blue Patch}

$l_{\bluebox} - 6 \le 6(1 - v_{6\bluebox})$ _{Column 6, Blue Patch}

$l_{\magentabox} - 1 \le 6(1 - v_{1\magentabox})$ _{Column 1, Magenta Patch}

$l_{\magentabox} - 2 \le 6(1 - v_{2\magentabox})$ _{Column 2, Magenta Patch}

$l_{\magentabox} - 3 \le 6(1 - v_{3\magentabox})$ _{Column 3, Magenta Patch}

$l_{\magentabox} - 4 \le 6(1 - v_{4\magentabox})$ _{Column 4, Magenta Patch}

$l_{\magentabox} - 5 \le 6(1 - v_{5\magentabox})$ _{Column 5, Magenta Patch}

$l_{\magentabox} - 6 \le 6(1 - v_{6\magentabox})$ _{Column 6, Magenta Patch}

$l_{\brickbox} - 1 \le 6(1 - v_{1\brickbox})$ _{Column 1, Brick Patch}

$l_{\brickbox} - 2 \le 6(1 - v_{2\brickbox})$ _{Column 2, Brick Patch}

$l_{\brickbox} - 3 \le 6(1 - v_{3\brickbox})$ _{Column 3, Brick Patch}

$l_{\brickbox} - 4 \le 6(1 - v_{4\brickbox})$ _{Column 4, Brick Patch}

$l_{\brickbox} - 5 \le 6(1 - v_{5\brickbox})$ _{Column 5, Brick Patch}

$l_{\brickbox} - 6 \le 6(1 - v_{6\brickbox})$ _{Column 6, Brick Patch}

$l_{\brownbox} - 1 \le 6(1 - v_{1\brownbox})$ _{Column 1, Brown Patch}

$l_{\brownbox} - 2 \le 6(1 - v_{2\brownbox})$ _{Column 2, Brown Patch}

$l_{\brownbox} - 3 \le 6(1 - v_{3\brownbox})$ _{Column 3, Brown Patch}

$l_{\brownbox} - 4 \le 6(1 - v_{4\brownbox})$ _{Column 4, Brown Patch}

$l_{\brownbox} - 5 \le 6(1 - v_{5\brownbox})$ _{Column 5, Brown Patch}

$l_{\brownbox} - 6 \le 6(1 - v_{6\brownbox})$ _{Column 6, Brown Patch}

Right Boundary Constraints

$1 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{1\yellowbox})$ _{Column 1, Yellow Patch}

$2 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{2\yellowbox})$ _{Column 2, Yellow Patch}

$3 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{3\yellowbox})$ _{Column 3, Yellow Patch}

$4 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{4\yellowbox})$ _{Column 4, Yellow Patch}

$5 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{5\yellowbox})$ _{Column 5, Yellow Patch}

$6 - (l_{\yellowbox} + w_{\yellowbox} - 1) \le 6(1 - v_{6\yellowbox})$ _{Column 6, Yellow Patch}

$1 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{1\tealbox})$ _{Column 1, Tealbox Patch}

$2 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{2\tealbox})$ _{Column 2, Tealbox Patch}

$3 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{3\tealbox})$ _{Column 3, Tealbox Patch}

$4 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{4\tealbox})$ _{Column 4, Tealbox Patch}

$5 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{5\tealbox})$ _{Column 5, Tealbox Patch}

$6 - (l_{\tealbox} + w_{\tealbox} - 1) \le 6(1 - v_{6\tealbox})$ _{Column 6, Tealbox Patch}

$1 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{1\purplebox})$ _{Column 1, Purple Patch}

$2 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{2\purplebox})$ _{Column 2, Purple Patch}

$3 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{3\purplebox})$ _{Column 3, Purple Patch}

$4 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{4\purplebox})$ _{Column 4, Purple Patch}

$5 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{5\purplebox})$ _{Column 5, Purple Patch}

$6 - (l_{\purplebox} + w_{\purplebox} - 1) \le 6(1 - v_{6\purplebox})$ _{Column 6, Purple Patch}

$1 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{1\greenbox})$ _{Column 1, Green Patch}

$2 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{2\greenbox})$ _{Column 2, Green Patch}

$3 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{3\greenbox})$ _{Column 3, Green Patch}

$4 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{4\greenbox})$ _{Column 4, Green Patch}

$5 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{5\greenbox})$ _{Column 5, Green Patch}

$6 - (l_{\greenbox} + w_{\greenbox} - 1) \le 6(1 - v_{6\greenbox})$ _{Column 6, Green Patch}

$1 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{1\orangebox})$ _{Column 1, Orange Patch}

$2 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{2\orangebox})$ _{Column 2, Orange Patch}

$3 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{3\orangebox})$ _{Column 3, Orange Patch}

$4 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{4\orangebox})$ _{Column 4, Orange Patch}

$5 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{5\orangebox})$ _{Column 5, Orange Patch}

$6 - (l_{\orangebox} + w_{\orangebox} - 1) \le 6(1 - v_{6\orangebox})$ _{Column 6, Orange Patch}

$1 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{1\redbox})$ _{Column 1, Red Patch}

$2 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{2\redbox})$ _{Column 2, Red Patch}

$3 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{3\redbox})$ _{Column 3, Red Patch}

$4 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{4\redbox})$ _{Column 4, Red Patch}

$5 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{5\redbox})$ _{Column 5, Red Patch}

$6 - (l_{\redbox} + w_{\redbox} - 1) \le 6(1 - v_{6\redbox})$ _{Column 6, Red Patch}

$1 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{1\bluebox})$ _{Column 1, Blue Patch}

$2 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{2\bluebox})$ _{Column 2, Blue Patch}

$3 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{3\bluebox})$ _{Column 3, Blue Patch}

$4 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{4\bluebox})$ _{Column 4, Blue Patch}

$5 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{5\bluebox})$ _{Column 5, Blue Patch}

$6 - (l_{\bluebox} + w_{\bluebox} - 1) \le 6(1 - v_{6\bluebox})$ _{Column 6, Blue Patch}

$1 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{1\magentabox})$ _{Column 1, Magenta Patch}

$2 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{2\magentabox})$ _{Column 2, Magenta Patch}

$3 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{3\magentabox})$ _{Column 3, Magenta Patch}

$4 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{4\magentabox})$ _{Column 4, Magenta Patch}

$5 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{5\magentabox})$ _{Column 5, Magenta Patch}

$6 - (l_{\magentabox} + w_{\magentabox} - 1) \le 6(1 - v_{6\magentabox})$ _{Column 6, Magenta Patch}

$1 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{1\brickbox})$ _{Column 1, Brick Patch}

$2 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{2\brickbox})$ _{Column 2, Brick Patch}

$3 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{3\brickbox})$ _{Column 3, Brick Patch}

$4 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{4\brickbox})$ _{Column 4, Brick Patch}

$5 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{5\brickbox})$ _{Column 5, Brick Patch}

$6 - (l_{\brickbox} + w_{\brickbox} - 1) \le 6(1 - v_{6\brickbox})$ _{Column 6, Brick Patch}

$1 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{1\brownbox})$ _{Column 1, Brown Patch}

$2 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{2\brownbox})$ _{Column 2, Brown Patch}

$3 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{3\brownbox})$ _{Column 3, Brown Patch}

$4 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{4\brownbox})$ _{Column 4, Brown Patch}

$5 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{5\brownbox})$ _{Column 5, Brown Patch}

$6 - (l_{\brownbox} + w_{\brownbox} - 1) \le 6(1 - v_{6\brownbox})$ _{Column 6, Brown Patch}

Height Constraints

$u_{1\yellowbox} + u_{2\yellowbox} + u_{3\yellowbox} + u_{4\yellowbox} + u_{5\yellowbox} + u_{6\yellowbox} = h_{\yellowbox}$ _{Yellow Patch_}

$u_{1\tealbox} + u_{2\tealbox} + u_{3\tealbox} + u_{4\tealbox} + u_{5\tealbox} + u_{6\tealbox} = h_{\tealbox}$ _{Teal Patch_}

$u_{1\purplebox} + u_{2\purplebox} + u_{3\purplebox} + u_{4\purplebox} + u_{5\purplebox} + u_{6\purplebox} = h_{\purplebox}$ _{Purple Patch_}

$u_{1\greenbox} + u_{2\greenbox} + u_{3\greenbox} + u_{4\greenbox} + u_{5\greenbox} + u_{6\greenbox} = h_{\greenbox}$ _{Green Patch_}

$u_{1\orangebox} + u_{2\orangebox} + u_{3\orangebox} + u_{4\orangebox} + u_{5\orangebox} + u_{6\orangebox} = h_{\orangebox}$ _{Orange Patch_}

$u_{1\redbox} + u_{2\redbox} + u_{3\redbox} + u_{4\redbox} + u_{5\redbox} + u_{6\redbox} = h_{\redbox}$ _{Red Patch_}

$u_{1\bluebox} + u_{2\bluebox} + u_{3\bluebox} + u_{4\bluebox} + u_{5\bluebox} + u_{6\bluebox} = h_{\bluebox}$ _{Blue Patch_}

$u_{1\magentabox} + u_{2\magentabox} + u_{3\magentabox} + u_{4\magentabox} + u_{5\magentabox} + u_{6\magentabox} = h_{\magentabox}$ _{Magenta Patch_}

$u_{1\brickbox} + u_{2\brickbox} + u_{3\brickbox} + u_{4\brickbox} + u_{5\brickbox} + u_{6\brickbox} = h_{\brickbox}$ _{Brick Patch_}

$u_{1\brownbox} + u_{2\brownbox} + u_{3\brownbox} + u_{4\brownbox} + u_{5\brownbox} + u_{6\brownbox} = h_{\brownbox}$ _{Brown Patch_}

Width Constraints

$v_{1\yellowbox} + v_{2\yellowbox} + v_{3\yellowbox} + v_{4\yellowbox} + v_{5\yellowbox} + v_{6\yellowbox} = w_{\yellowbox}$ _{Yellow Patch_}

$v_{1\tealbox} + v_{2\tealbox} + v_{3\tealbox} + v_{4\tealbox} + v_{5\tealbox} + v_{6\tealbox} = w_{\tealbox}$ _{Teal Patch_}

$v_{1\purplebox} + v_{2\purplebox} + v_{3\purplebox} + v_{4\purplebox} + v_{5\purplebox} + v_{6\purplebox} = w_{\purplebox}$ _{Purple Patch_}

$v_{1\greenbox} + v_{2\greenbox} + v_{3\greenbox} + v_{4\greenbox} + v_{5\greenbox} + v_{6\greenbox} = w_{\greenbox}$ _{Green Patch_}

$v_{1\orangebox} + v_{2\orangebox} + v_{3\orangebox} + v_{4\orangebox} + v_{5\orangebox} + v_{6\orangebox} = w_{\orangebox}$ _{Orange Patch_}

$v_{1\redbox} + v_{2\redbox} + v_{3\redbox} + v_{4\redbox} + v_{5\redbox} + v_{6\redbox} = w_{\redbox}$ _{Red Patch_}

$v_{1\bluebox} + v_{2\bluebox} + v_{3\bluebox} + v_{4\bluebox} + v_{5\bluebox} + v_{6\bluebox} = w_{\bluebox}$ _{Blue Patch_}

$v_{1\magentabox} + v_{2\magentabox} + v_{3\magentabox} + v_{4\magentabox} + v_{5\magentabox} + v_{6\magentabox} = w_{\magentabox}$ _{Magenta Patch_}

$v_{1\brickbox} + v_{2\brickbox} + v_{3\brickbox} + v_{4\brickbox} + v_{5\brickbox} + v_{6\brickbox} = w_{\brickbox}$ _{Brick Patch_}

$v_{1\brownbox} + v_{2\brownbox} + v_{3\brownbox} + v_{4\brownbox} + v_{5\brownbox} + v_{6\brownbox} = w_{\brownbox}$ _{Brown Patch_}

Cutout-Rows Constraints

$x_{11\yellowbox} + x_{12\yellowbox} + x_{13\yellowbox} + x_{14\yellowbox} + x_{15\yellowbox} + x_{16\yellowbox} \le 6u_{1\yellowbox}$ _{Row 1, Yellow Patch}

$x_{21\yellowbox} + x_{22\yellowbox} + x_{23\yellowbox} + x_{24\yellowbox} + x_{25\yellowbox} + x_{26\yellowbox} \le 6u_{2\yellowbox}$ _{Row 2, Yellow Patch}

$x_{31\yellowbox} + x_{32\yellowbox} + x_{33\yellowbox} + x_{34\yellowbox} + x_{35\yellowbox} + x_{36\yellowbox} \le 6u_{3\yellowbox}$ _{Row 3, Yellow Patch}

$x_{41\yellowbox} + x_{42\yellowbox} + x_{43\yellowbox} + x_{44\yellowbox} + x_{45\yellowbox} + x_{46\yellowbox} \le 6u_{4\yellowbox}$ _{Row 4, Yellow Patch}

$x_{51\yellowbox} + x_{52\yellowbox} + x_{53\yellowbox} + x_{54\yellowbox} + x_{55\yellowbox} + x_{56\yellowbox} \le 6u_{5\yellowbox}$ _{Row 5, Yellow Patch}

$x_{61\yellowbox} + x_{62\yellowbox} + x_{63\yellowbox} + x_{64\yellowbox} + x_{65\yellowbox} + x_{66\yellowbox} \le 6u_{6\yellowbox}$ _{Row 6, Yellow Patch}

$x_{11\tealbox} + x_{12\tealbox} + x_{13\tealbox} + x_{14\tealbox} + x_{15\tealbox} + x_{16\tealbox} \le 6u_{1\tealbox}$ _{Row 1, Teal}

$x_{21\tealbox} + x_{22\tealbox} + x_{23\tealbox} + x_{24\tealbox} + x_{25\tealbox} + x_{26\tealbox} \le 6u_{2\tealbox}$ _{Row 2, Teal}

$x_{31\tealbox} + x_{32\tealbox} + x_{33\tealbox} + x_{34\tealbox} + x_{35\tealbox} + x_{36\tealbox} \le 6u_{3\tealbox}$ _{Row 3, Teal}

$x_{41\tealbox} + x_{42\tealbox} + x_{43\tealbox} + x_{44\tealbox} + x_{45\tealbox} + x_{46\tealbox} \le 6u_{4\tealbox}$ _{Row 4, Teal}

$x_{51\tealbox} + x_{52\tealbox} + x_{53\tealbox} + x_{54\tealbox} + x_{55\tealbox} + x_{56\tealbox} \le 6u_{5\tealbox}$ _{Row 5, Teal}

$x_{61\tealbox} + x_{62\tealbox} + x_{63\tealbox} + x_{64\tealbox} + x_{65\tealbox} + x_{66\tealbox} \le 6u_{6\tealbox}$ _{Row 6, Teal}

$x_{11\purplebox} + x_{12\purplebox} + x_{13\purplebox} + x_{14\purplebox} + x_{15\purplebox} + x_{16\purplebox} \le 6u_{1\purplebox}$ _{Row 1, Purple Patch}

$x_{21\purplebox} + x_{22\purplebox} + x_{23\purplebox} + x_{24\purplebox} + x_{25\purplebox} + x_{26\purplebox} \le 6u_{2\purplebox}$ _{Row 2, Purple Patch}

$x_{31\purplebox} + x_{32\purplebox} + x_{33\purplebox} + x_{34\purplebox} + x_{35\purplebox} + x_{36\purplebox} \le 6u_{3\purplebox}$ _{Row 3, Purple Patch}

$x_{41\purplebox} + x_{42\purplebox} + x_{43\purplebox} + x_{44\purplebox} + x_{45\purplebox} + x_{46\purplebox} \le 6u_{4\purplebox}$ _{Row 4, Purple Patch}

$x_{51\purplebox} + x_{52\purplebox} + x_{53\purplebox} + x_{54\purplebox} + x_{55\purplebox} + x_{56\purplebox} \le 6u_{5\purplebox}$ _{Row 5, Purple Patch}

$x_{61\purplebox} + x_{62\purplebox} + x_{63\purplebox} + x_{64\purplebox} + x_{65\purplebox} + x_{66\purplebox} \le 6u_{6\purplebox}$ _{Row 6, Purple Patch}

$x_{11\greenbox} + x_{12\greenbox} + x_{13\greenbox} + x_{14\greenbox} + x_{15\greenbox} + x_{16\greenbox} \le 6u_{1\greenbox}$ _{Row 1, Green Patch}

$x_{21\greenbox} + x_{22\greenbox} + x_{23\greenbox} + x_{24\greenbox} + x_{25\greenbox} + x_{26\greenbox} \le 6u_{2\greenbox}$ _{Row 2, Green Patch}

$x_{31\greenbox} + x_{32\greenbox} + x_{33\greenbox} + x_{34\greenbox} + x_{35\greenbox} + x_{36\greenbox} \le 6u_{3\greenbox}$ _{Row 3, Green Patch}

$x_{41\greenbox} + x_{42\greenbox} + x_{43\greenbox} + x_{44\greenbox} + x_{45\greenbox} + x_{46\greenbox} \le 6u_{4\greenbox}$ _{Row 4, Green Patch}

$x_{51\greenbox} + x_{52\greenbox} + x_{53\greenbox} + x_{54\greenbox} + x_{55\greenbox} + x_{56\greenbox} \le 6u_{5\greenbox}$ _{Row 5, Green Patch}

$x_{61\greenbox} + x_{62\greenbox} + x_{63\greenbox} + x_{64\greenbox} + x_{65\greenbox} + x_{66\greenbox} \le 6u_{6\greenbox}$ _{Row 6, Green Patch}

$x_{11\orangebox} + x_{12\orangebox} + x_{13\orangebox} + x_{14\orangebox} + x_{15\orangebox} + x_{16\orangebox} \le 6u_{1\orangebox}$ _{Row 1, Orange Patch}

$x_{21\orangebox} + x_{22\orangebox} + x_{23\orangebox} + x_{24\orangebox} + x_{25\orangebox} + x_{26\orangebox} \le 6u_{2\orangebox}$ _{Row 2, Orange Patch}

$x_{31\orangebox} + x_{32\orangebox} + x_{33\orangebox} + x_{34\orangebox} + x_{35\orangebox} + x_{36\orangebox} \le 6u_{3\orangebox}$ _{Row 3, Orange Patch}

$x_{41\orangebox} + x_{42\orangebox} + x_{43\orangebox} + x_{44\orangebox} + x_{45\orangebox} + x_{46\orangebox} \le 6u_{4\orangebox}$ _{Row 4, Orange Patch}

$x_{51\orangebox} + x_{52\orangebox} + x_{53\orangebox} + x_{54\orangebox} + x_{55\orangebox} + x_{56\orangebox} \le 6u_{5\orangebox}$ _{Row 5, Orange Patch}

$x_{61\orangebox} + x_{62\orangebox} + x_{63\orangebox} + x_{64\orangebox} + x_{65\orangebox} + x_{66\orangebox} \le 6u_{6\orangebox}$ _{Row 6, Orange Patch}

$x_{11\redbox} + x_{12\redbox} + x_{13\redbox} + x_{14\redbox} + x_{15\redbox} + x_{16\redbox} \le 6u_{1\redbox}$ _{Row 1, Red Patch}

$x_{21\redbox} + x_{22\redbox} + x_{23\redbox} + x_{24\redbox} + x_{25\redbox} + x_{26\redbox} \le 6u_{2\redbox}$ _{Row 2, Red Patch}

$x_{31\redbox} + x_{32\redbox} + x_{33\redbox} + x_{34\redbox} + x_{35\redbox} + x_{36\redbox} \le 6u_{3\redbox}$ _{Row 3, Red Patch}

$x_{41\redbox} + x_{42\redbox} + x_{43\redbox} + x_{44\redbox} + x_{45\redbox} + x_{46\redbox} \le 6u_{4\redbox}$ _{Row 4, Red Patch}

$x_{51\redbox} + x_{52\redbox} + x_{53\redbox} + x_{54\redbox} + x_{55\redbox} + x_{56\redbox} \le 6u_{5\redbox}$ _{Row 5, Red Patch}

$x_{61\redbox} + x_{62\redbox} + x_{63\redbox} + x_{64\redbox} + x_{65\redbox} + x_{66\redbox} \le 6u_{6\redbox}$ _{Row 6, Red Patch}

$x_{11\bluebox} + x_{12\bluebox} + x_{13\bluebox} + x_{14\bluebox} + x_{15\bluebox} + x_{16\bluebox} \le 6u_{1\bluebox}$ _{Row 1, Blue Patch}

$x_{21\bluebox} + x_{22\bluebox} + x_{23\bluebox} + x_{24\bluebox} + x_{25\bluebox} + x_{26\bluebox} \le 6u_{2\bluebox}$ _{Row 2, Blue Patch}

$x_{31\bluebox} + x_{32\bluebox} + x_{33\bluebox} + x_{34\bluebox} + x_{35\bluebox} + x_{36\bluebox} \le 6u_{3\bluebox}$ _{Row 3, Blue Patch}

$x_{41\bluebox} + x_{42\bluebox} + x_{43\bluebox} + x_{44\bluebox} + x_{45\bluebox} + x_{46\bluebox} \le 6u_{4\bluebox}$ _{Row 4, Blue Patch}

$x_{51\bluebox} + x_{52\bluebox} + x_{53\bluebox} + x_{54\bluebox} + x_{55\bluebox} + x_{56\bluebox} \le 6u_{5\bluebox}$ _{Row 5, Blue Patch}

$x_{61\bluebox} + x_{62\bluebox} + x_{63\bluebox} + x_{64\bluebox} + x_{65\bluebox} + x_{66\bluebox} \le 6u_{6\bluebox}$ _{Row 6, Blue Patch}

$x_{11\magentabox} + x_{12\magentabox} + x_{13\magentabox} + x_{14\magentabox} + x_{15\magentabox} + x_{16\magentabox} \le 6u_{1\magentabox}$ _{Row 1, Magenta Patch}

$x_{21\magentabox} + x_{22\magentabox} + x_{23\magentabox} + x_{24\magentabox} + x_{25\magentabox} + x_{26\magentabox} \le 6u_{2\magentabox}$ _{Row 2, Magenta Patch}

$x_{31\magentabox} + x_{32\magentabox} + x_{33\magentabox} + x_{34\magentabox} + x_{35\magentabox} + x_{36\magentabox} \le 6u_{3\magentabox}$ _{Row 3, Magenta Patch}

$x_{41\magentabox} + x_{42\magentabox} + x_{43\magentabox} + x_{44\magentabox} + x_{45\magentabox} + x_{46\magentabox} \le 6u_{4\magentabox}$ _{Row 4, Magenta Patch}

$x_{51\magentabox} + x_{52\magentabox} + x_{53\magentabox} + x_{54\magentabox} + x_{55\magentabox} + x_{56\magentabox} \le 6u_{5\magentabox}$ _{Row 5, Magenta Patch}

$x_{61\magentabox} + x_{62\magentabox} + x_{63\magentabox} + x_{64\magentabox} + x_{65\magentabox} + x_{66\magentabox} \le 6u_{6\magentabox}$ _{Row 6, Magenta Patch}

$x_{11\brickbox} + x_{12\brickbox} + x_{13\brickbox} + x_{14\brickbox} + x_{15\brickbox} + x_{16\brickbox} \le 6u_{1\brickbox}$ _{Row 1, Brick Patch}

$x_{21\brickbox} + x_{22\brickbox} + x_{23\brickbox} + x_{24\brickbox} + x_{25\brickbox} + x_{26\brickbox} \le 6u_{2\brickbox}$ _{Row 2, Brick Patch}

$x_{31\brickbox} + x_{32\brickbox} + x_{33\brickbox} + x_{34\brickbox} + x_{35\brickbox} + x_{36\brickbox} \le 6u_{3\brickbox}$ _{Row 3, Brick Patch}

$x_{41\brickbox} + x_{42\brickbox} + x_{43\brickbox} + x_{44\brickbox} + x_{45\brickbox} + x_{46\brickbox} \le 6u_{4\brickbox}$ _{Row 4, Brick Patch}

$x_{51\brickbox} + x_{52\brickbox} + x_{53\brickbox} + x_{54\brickbox} + x_{55\brickbox} + x_{56\brickbox} \le 6u_{5\brickbox}$ _{Row 5, Brick Patch}

$x_{61\brickbox} + x_{62\brickbox} + x_{63\brickbox} + x_{64\brickbox} + x_{65\brickbox} + x_{66\brickbox} \le 6u_{6\brickbox}$ _{Row 6, Brick Patch}

$x_{11\brownbox} + x_{12\brownbox} + x_{13\brownbox} + x_{14\brownbox} + x_{15\brownbox} + x_{16\brownbox} \le 6u_{1\brownbox}$ _{Row 1, Brown Patch}

$x_{21\brownbox} + x_{22\brownbox} + x_{23\brownbox} + x_{24\brownbox} + x_{25\brownbox} + x_{26\brownbox} \le 6u_{2\brownbox}$ _{Row 2, Brown Patch}

$x_{31\brownbox} + x_{32\brownbox} + x_{33\brownbox} + x_{34\brownbox} + x_{35\brownbox} + x_{36\brownbox} \le 6u_{3\brownbox}$ _{Row 3, Brown Patch}

$x_{41\brownbox} + x_{42\brownbox} + x_{43\brownbox} + x_{44\brownbox} + x_{45\brownbox} + x_{46\brownbox} \le 6u_{4\brownbox}$ _{Row 4, Brown Patch}

$x_{51\brownbox} + x_{52\brownbox} + x_{53\brownbox} + x_{54\brownbox} + x_{55\brownbox} + x_{56\brownbox} \le 6u_{5\brownbox}$ _{Row 5, Brown Patch}

$x_{61\brownbox} + x_{62\brownbox} + x_{63\brownbox} + x_{64\brownbox} + x_{65\brownbox} + x_{66\brownbox} \le 6u_{6\brownbox}$ _{Row 6, Brown Patch}

Cutout-Columns Constraints

$x_{11\yellowbox} + x_{21\yellowbox} + x_{31\yellowbox} + x_{41\yellowbox} + x_{51\yellowbox} + x_{61\yellowbox} \le 6v_{1\yellowbox}$ _{Column 1, Yellow Patch}

$x_{12\yellowbox} + x_{22\yellowbox} + x_{32\yellowbox} + x_{42\yellowbox} + x_{52\yellowbox} + x_{62\yellowbox} \le 6v_{2\yellowbox}$ _{Column 2, Yellow Patch}

$x_{13\yellowbox} + x_{23\yellowbox} + x_{33\yellowbox} + x_{43\yellowbox} + x_{53\yellowbox} + x_{63\yellowbox} \le 6v_{3\yellowbox}$ _{Column 3, Yellow Patch}

$x_{14\yellowbox} + x_{24\yellowbox} + x_{34\yellowbox} + x_{44\yellowbox} + x_{54\yellowbox} + x_{64\yellowbox} \le 6v_{4\yellowbox}$ _{Column 4, Yellow Patch}

$x_{15\yellowbox} + x_{25\yellowbox} + x_{35\yellowbox} + x_{45\yellowbox} + x_{55\yellowbox} + x_{65\yellowbox} \le 6v_{5\yellowbox}$ _{Column 5, Yellow Patch}

$x_{16\yellowbox} + x_{26\yellowbox} + x_{36\yellowbox} + x_{46\yellowbox} + x_{56\yellowbox} + x_{66\yellowbox} \le 6v_{6\yellowbox}$ _{Column 6, Yellow Patch}

$x_{11\tealbox} + x_{21\tealbox} + x_{31\tealbox} + x_{41\tealbox} + x_{51\tealbox} + x_{61\tealbox} \le 6v_{1\tealbox}$ _{Column 1, Teal}

$x_{12\tealbox} + x_{22\tealbox} + x_{32\tealbox} + x_{42\tealbox} + x_{52\tealbox} + x_{62\tealbox} \le 6v_{2\tealbox}$ _{Column 2, Teal}

$x_{13\tealbox} + x_{23\tealbox} + x_{33\tealbox} + x_{43\tealbox} + x_{53\tealbox} + x_{63\tealbox} \le 6v_{3\tealbox}$ _{Column 3, Teal}

$x_{14\tealbox} + x_{24\tealbox} + x_{34\tealbox} + x_{44\tealbox} + x_{54\tealbox} + x_{64\tealbox} \le 6v_{4\tealbox}$ _{Column 4, Teal}

$x_{15\tealbox} + x_{25\tealbox} + x_{35\tealbox} + x_{45\tealbox} + x_{55\tealbox} + x_{65\tealbox} \le 6v_{5\tealbox}$ _{Column 5, Teal}

$x_{16\tealbox} + x_{26\tealbox} + x_{36\tealbox} + x_{46\tealbox} + x_{56\tealbox} + x_{66\tealbox} \le 6v_{6\tealbox}$ _{Column 6, Teal}

$x_{11\purplebox} + x_{21\purplebox} + x_{31\purplebox} + x_{41\purplebox} + x_{51\purplebox} + x_{61\purplebox} \le 6v_{1\purplebox}$ _{Column 1, Purple Patch}

$x_{12\purplebox} + x_{22\purplebox} + x_{32\purplebox} + x_{42\purplebox} + x_{52\purplebox} + x_{62\purplebox} \le 6v_{2\purplebox}$ _{Column 2, Purple Patch}

$x_{13\purplebox} + x_{23\purplebox} + x_{33\purplebox} + x_{43\purplebox} + x_{53\purplebox} + x_{63\purplebox} \le 6v_{3\purplebox}$ _{Column 3, Purple Patch}

$x_{14\purplebox} + x_{24\purplebox} + x_{34\purplebox} + x_{44\purplebox} + x_{54\purplebox} + x_{64\purplebox} \le 6v_{4\purplebox}$ _{Column 4, Purple Patch}

$x_{15\purplebox} + x_{25\purplebox} + x_{35\purplebox} + x_{45\purplebox} + x_{55\purplebox} + x_{65\purplebox} \le 6v_{5\purplebox}$ _{Column 5, Purple Patch}

$x_{16\purplebox} + x_{26\purplebox} + x_{36\purplebox} + x_{46\purplebox} + x_{56\purplebox} + x_{66\purplebox} \le 6v_{6\purplebox}$ _{Column 6, Purple Patch}

$x_{11\greenbox} + x_{21\greenbox} + x_{31\greenbox} + x_{41\greenbox} + x_{51\greenbox} + x_{61\greenbox} \le 6v_{1\greenbox}$ _{Column 1, Green Patch}

$x_{12\greenbox} + x_{22\greenbox} + x_{32\greenbox} + x_{42\greenbox} + x_{52\greenbox} + x_{62\greenbox} \le 6v_{2\greenbox}$ _{Column 2, Green Patch}

$x_{13\greenbox} + x_{23\greenbox} + x_{33\greenbox} + x_{43\greenbox} + x_{53\greenbox} + x_{63\greenbox} \le 6v_{3\greenbox}$ _{Column 3, Green Patch}

$x_{14\greenbox} + x_{24\greenbox} + x_{34\greenbox} + x_{44\greenbox} + x_{54\greenbox} + x_{64\greenbox} \le 6v_{4\greenbox}$ _{Column 4, Green Patch}

$x_{15\greenbox} + x_{25\greenbox} + x_{35\greenbox} + x_{45\greenbox} + x_{55\greenbox} + x_{65\greenbox} \le 6v_{5\greenbox}$ _{Column 5, Green Patch}

$x_{16\greenbox} + x_{26\greenbox} + x_{36\greenbox} + x_{46\greenbox} + x_{56\greenbox} + x_{66\greenbox} \le 6v_{6\greenbox}$ _{Column 6, Green Patch}

$x_{11\orangebox} + x_{21\orangebox} + x_{31\orangebox} + x_{41\orangebox} + x_{51\orangebox} + x_{61\orangebox} \le 6v_{1\orangebox}$ _{Column 1, Orange Patch}

$x_{12\orangebox} + x_{22\orangebox} + x_{32\orangebox} + x_{42\orangebox} + x_{52\orangebox} + x_{62\orangebox} \le 6v_{2\orangebox}$ _{Column 2, Orange Patch}

$x_{13\orangebox} + x_{23\orangebox} + x_{33\orangebox} + x_{43\orangebox} + x_{53\orangebox} + x_{63\orangebox} \le 6v_{3\orangebox}$ _{Column 3, Orange Patch}

$x_{14\orangebox} + x_{24\orangebox} + x_{34\orangebox} + x_{44\orangebox} + x_{54\orangebox} + x_{64\orangebox} \le 6v_{4\orangebox}$ _{Column 4, Orange Patch}

$x_{15\orangebox} + x_{25\orangebox} + x_{35\orangebox} + x_{45\orangebox} + x_{55\orangebox} + x_{65\orangebox} \le 6v_{5\orangebox}$ _{Column 5, Orange Patch}

$x_{16\orangebox} + x_{26\orangebox} + x_{36\orangebox} + x_{46\orangebox} + x_{56\orangebox} + x_{66\orangebox} \le 6v_{6\orangebox}$ _{Column 6, Orange Patch}

$x_{11\redbox} + x_{21\redbox} + x_{31\redbox} + x_{41\redbox} + x_{51\redbox} + x_{61\redbox} \le 6v_{1\redbox}$ _{Column 1, Red Patch}

$x_{12\redbox} + x_{22\redbox} + x_{32\redbox} + x_{42\redbox} + x_{52\redbox} + x_{62\redbox} \le 6v_{2\redbox}$ _{Column 2, Red Patch}

$x_{13\redbox} + x_{23\redbox} + x_{33\redbox} + x_{43\redbox} + x_{53\redbox} + x_{63\redbox} \le 6v_{3\redbox}$ _{Column 3, Red Patch}

$x_{14\redbox} + x_{24\redbox} + x_{34\redbox} + x_{44\redbox} + x_{54\redbox} + x_{64\redbox} \le 6v_{4\redbox}$ _{Column 4, Red Patch}

$x_{15\redbox} + x_{25\redbox} + x_{35\redbox} + x_{45\redbox} + x_{55\redbox} + x_{65\redbox} \le 6v_{5\redbox}$ _{Column 5, Red Patch}

$x_{16\redbox} + x_{26\redbox} + x_{36\redbox} + x_{46\redbox} + x_{56\redbox} + x_{66\redbox} \le 6v_{6\redbox}$ _{Column 6, Red Patch}

$x_{11\bluebox} + x_{21\bluebox} + x_{31\bluebox} + x_{41\bluebox} + x_{51\bluebox} + x_{61\bluebox} \le 6v_{1\bluebox}$ _{Column 1, Blue Patch}

$x_{12\bluebox} + x_{22\bluebox} + x_{32\bluebox} + x_{42\bluebox} + x_{52\bluebox} + x_{62\bluebox} \le 6v_{2\bluebox}$ _{Column 2, Blue Patch}

$x_{13\bluebox} + x_{23\bluebox} + x_{33\bluebox} + x_{43\bluebox} + x_{53\bluebox} + x_{63\bluebox} \le 6v_{3\bluebox}$ _{Column 3, Blue Patch}

$x_{14\bluebox} + x_{24\bluebox} + x_{34\bluebox} + x_{44\bluebox} + x_{54\bluebox} + x_{64\bluebox} \le 6v_{4\bluebox}$ _{Column 4, Blue Patch}

$x_{15\bluebox} + x_{25\bluebox} + x_{35\bluebox} + x_{45\bluebox} + x_{55\bluebox} + x_{65\bluebox} \le 6v_{5\bluebox}$ _{Column 5, Blue Patch}

$x_{16\bluebox} + x_{26\bluebox} + x_{36\bluebox} + x_{46\bluebox} + x_{56\bluebox} + x_{66\bluebox} \le 6v_{6\bluebox}$ _{Column 6, Blue Patch}

$x_{11\magentabox} + x_{21\magentabox} + x_{31\magentabox} + x_{41\magentabox} + x_{51\magentabox} + x_{61\magentabox} \le 6v_{1\magentabox}$ _{Column 1, Magenta Patch}

$x_{12\magentabox} + x_{22\magentabox} + x_{32\magentabox} + x_{42\magentabox} + x_{52\magentabox} + x_{62\magentabox} \le 6v_{2\magentabox}$ _{Column 2, Magenta Patch}

$x_{13\magentabox} + x_{23\magentabox} + x_{33\magentabox} + x_{43\magentabox} + x_{53\magentabox} + x_{63\magentabox} \le 6v_{3\magentabox}$ _{Column 3, Magenta Patch}

$x_{14\magentabox} + x_{24\magentabox} + x_{34\magentabox} + x_{44\magentabox} + x_{54\magentabox} + x_{64\magentabox} \le 6v_{4\magentabox}$ _{Column 4, Magenta Patch}

$x_{15\magentabox} + x_{25\magentabox} + x_{35\magentabox} + x_{45\magentabox} + x_{55\magentabox} + x_{65\magentabox} \le 6v_{5\magentabox}$ _{Column 5, Magenta Patch}

$x_{16\magentabox} + x_{26\magentabox} + x_{36\magentabox} + x_{46\magentabox} + x_{56\magentabox} + x_{66\magentabox} \le 6v_{6\magentabox}$ _{Column 6, Magenta Patch}

$x_{11\brickbox} + x_{21\brickbox} + x_{31\brickbox} + x_{41\brickbox} + x_{51\brickbox} + x_{61\brickbox} \le 6v_{1\brickbox}$ _{Column 1, Brick Patch}

$x_{12\brickbox} + x_{22\brickbox} + x_{32\brickbox} + x_{42\brickbox} + x_{52\brickbox} + x_{62\brickbox} \le 6v_{2\brickbox}$ _{Column 2, Brick Patch}

$x_{13\brickbox} + x_{23\brickbox} + x_{33\brickbox} + x_{43\brickbox} + x_{53\brickbox} + x_{63\brickbox} \le 6v_{3\brickbox}$ _{Column 3, Brick Patch}

$x_{14\brickbox} + x_{24\brickbox} + x_{34\brickbox} + x_{44\brickbox} + x_{54\brickbox} + x_{64\brickbox} \le 6v_{4\brickbox}$ _{Column 4, Brick Patch}

$x_{15\brickbox} + x_{25\brickbox} + x_{35\brickbox} + x_{45\brickbox} + x_{55\brickbox} + x_{65\brickbox} \le 6v_{5\brickbox}$ _{Column 5, Brick Patch}

$x_{16\brickbox} + x_{26\brickbox} + x_{36\brickbox} + x_{46\brickbox} + x_{56\brickbox} + x_{66\brickbox} \le 6v_{6\brickbox}$ _{Column 6, Brick Patch}

$x_{11\brownbox} + x_{21\brownbox} + x_{31\brownbox} + x_{41\brownbox} + x_{51\brownbox} + x_{61\brownbox} \le 6v_{1\brownbox}$ _{Column 1, Brown Patch}

$x_{12\brownbox} + x_{22\brownbox} + x_{32\brownbox} + x_{42\brownbox} + x_{52\brownbox} + x_{62\brownbox} \le 6v_{2\brownbox}$ _{Column 2, Brown Patch}

$x_{13\brownbox} + x_{23\brownbox} + x_{33\brownbox} + x_{43\brownbox} + x_{53\brownbox} + x_{63\brownbox} \le 6v_{3\brownbox}$ _{Column 3, Brown Patch}

$x_{14\brownbox} + x_{24\brownbox} + x_{34\brownbox} + x_{44\brownbox} + x_{54\brownbox} + x_{64\brownbox} \le 6v_{4\brownbox}$ _{Column 4, Brown Patch}

$x_{15\brownbox} + x_{25\brownbox} + x_{35\brownbox} + x_{45\brownbox} + x_{55\brownbox} + x_{65\brownbox} \le 6v_{5\brownbox}$ _{Column 5, Brown Patch}

$x_{16\brownbox} + x_{26\brownbox} + x_{36\brownbox} + x_{46\brownbox} + x_{56\brownbox} + x_{66\brownbox} \le 6v_{6\brownbox}$ _{Column 6, Brown Patch}

Square Coverage Constraints

$x_{11\yellowbox} \ge u_{1\yellowbox} + v_{1\yellowbox} - 1$ _{Square (1, 1), Yellow Patch}

$x_{12\yellowbox} \ge u_{1\yellowbox} + v_{2\yellowbox} - 1$ _{Square (1, 2), Yellow Patch}

$x_{13\yellowbox} \ge u_{1\yellowbox} + v_{3\yellowbox} - 1$ _{Square (1, 3), Yellow Patch}

$x_{14\yellowbox} \ge u_{1\yellowbox} + v_{4\yellowbox} - 1$ _{Square (1, 4), Yellow Patch}

$x_{15\yellowbox} \ge u_{1\yellowbox} + v_{5\yellowbox} - 1$ _{Square (1, 5), Yellow Patch}

$x_{16\yellowbox} \ge u_{1\yellowbox} + v_{6\yellowbox} - 1$ _{Square (1, 6), Yellow Patch}

$x_{21\yellowbox} \ge u_{2\yellowbox} + v_{1\yellowbox} - 1$ _{Square (2, 1), Yellow Patch}

$x_{22\yellowbox} \ge u_{2\yellowbox} + v_{2\yellowbox} - 1$ _{Square (2, 2), Yellow Patch}

$x_{23\yellowbox} \ge u_{2\yellowbox} + v_{3\yellowbox} - 1$ _{Square (2, 3), Yellow Patch}

$x_{24\yellowbox} \ge u_{2\yellowbox} + v_{4\yellowbox} - 1$ _{Square (2, 4), Yellow Patch}

$x_{25\yellowbox} \ge u_{2\yellowbox} + v_{5\yellowbox} - 1$ _{Square (2, 5), Yellow Patch}

$x_{26\yellowbox} \ge u_{2\yellowbox} + v_{6\yellowbox} - 1$ _{Square (2, 6), Yellow Patch}

$x_{31\yellowbox} \ge u_{3\yellowbox} + v_{1\yellowbox} - 1$ _{Square (3, 1), Yellow Patch}

$x_{32\yellowbox} \ge u_{3\yellowbox} + v_{2\yellowbox} - 1$ _{Square (3, 2), Yellow Patch}

$x_{33\yellowbox} \ge u_{3\yellowbox} + v_{3\yellowbox} - 1$ _{Square (3, 3), Yellow Patch}

$x_{34\yellowbox} \ge u_{3\yellowbox} + v_{4\yellowbox} - 1$ _{Square (3, 4), Yellow Patch}

$x_{35\yellowbox} \ge u_{3\yellowbox} + v_{5\yellowbox} - 1$ _{Square (3, 5), Yellow Patch}

$x_{36\yellowbox} \ge u_{3\yellowbox} + v_{6\yellowbox} - 1$ _{Square (3, 6), Yellow Patch}

$x_{41\yellowbox} \ge u_{4\yellowbox} + v_{1\yellowbox} - 1$ _{Square (4, 1), Yellow Patch}

$x_{42\yellowbox} \ge u_{4\yellowbox} + v_{2\yellowbox} - 1$ _{Square (4, 2), Yellow Patch}

$x_{43\yellowbox} \ge u_{4\yellowbox} + v_{3\yellowbox} - 1$ _{Square (4, 3), Yellow Patch}

$x_{44\yellowbox} \ge u_{4\yellowbox} + v_{4\yellowbox} - 1$ _{Square (4, 4), Yellow Patch}

$x_{45\yellowbox} \ge u_{4\yellowbox} + v_{5\yellowbox} - 1$ _{Square (4, 5), Yellow Patch}

$x_{46\yellowbox} \ge u_{4\yellowbox} + v_{6\yellowbox} - 1$ _{Square (4, 6), Yellow Patch}

$x_{51\yellowbox} \ge u_{5\yellowbox} + v_{1\yellowbox} - 1$ _{Square (5, 1), Yellow Patch}

$x_{52\yellowbox} \ge u_{5\yellowbox} + v_{2\yellowbox} - 1$ _{Square (5, 2), Yellow Patch}

$x_{53\yellowbox} \ge u_{5\yellowbox} + v_{3\yellowbox} - 1$ _{Square (5, 3), Yellow Patch}

$x_{54\yellowbox} \ge u_{5\yellowbox} + v_{4\yellowbox} - 1$ _{Square (5, 4), Yellow Patch}

$x_{55\yellowbox} \ge u_{5\yellowbox} + v_{5\yellowbox} - 1$ _{Square (5, 5), Yellow Patch}

$x_{56\yellowbox} \ge u_{5\yellowbox} + v_{6\yellowbox} - 1$ _{Square (5, 6), Yellow Patch}

$x_{61\yellowbox} \ge u_{6\yellowbox} + v_{1\yellowbox} - 1$ _{Square (6, 1), Yellow Patch}

$x_{62\yellowbox} \ge u_{6\yellowbox} + v_{2\yellowbox} - 1$ _{Square (6, 2), Yellow Patch}

$x_{63\yellowbox} \ge u_{6\yellowbox} + v_{3\yellowbox} - 1$ _{Square (6, 3), Yellow Patch}

$x_{64\yellowbox} \ge u_{6\yellowbox} + v_{4\yellowbox} - 1$ _{Square (6, 4), Yellow Patch}

$x_{65\yellowbox} \ge u_{6\yellowbox} + v_{5\yellowbox} - 1$ _{Square (6, 5), Yellow Patch}

$x_{66\yellowbox} \ge u_{6\yellowbox} + v_{6\yellowbox} - 1$ _{Square (6, 6), Yellow Patch}

$x_{11\tealbox} \ge u_{1\tealbox} + v_{1\tealbox} - 1$ _{Square (1, 1), Tealbox Patch}

$x_{12\tealbox} \ge u_{1\tealbox} + v_{2\tealbox} - 1$ _{Square (1, 2), Tealbox Patch}

$x_{13\tealbox} \ge u_{1\tealbox} + v_{3\tealbox} - 1$ _{Square (1, 3), Tealbox Patch}

$x_{14\tealbox} \ge u_{1\tealbox} + v_{4\tealbox} - 1$ _{Square (1, 4), Tealbox Patch}

$x_{15\tealbox} \ge u_{1\tealbox} + v_{5\tealbox} - 1$ _{Square (1, 5), Tealbox Patch}

$x_{16\tealbox} \ge u_{1\tealbox} + v_{6\tealbox} - 1$ _{Square (1, 6), Tealbox Patch}

$x_{21\tealbox} \ge u_{2\tealbox} + v_{1\tealbox} - 1$ _{Square (2, 1), Tealbox Patch}

$x_{22\tealbox} \ge u_{2\tealbox} + v_{2\tealbox} - 1$ _{Square (2, 2), Tealbox Patch}

$x_{23\tealbox} \ge u_{2\tealbox} + v_{3\tealbox} - 1$ _{Square (2, 3), Tealbox Patch}

$x_{24\tealbox} \ge u_{2\tealbox} + v_{4\tealbox} - 1$ _{Square (2, 4), Tealbox Patch}

$x_{25\tealbox} \ge u_{2\tealbox} + v_{5\tealbox} - 1$ _{Square (2, 5), Tealbox Patch}

$x_{26\tealbox} \ge u_{2\tealbox} + v_{6\tealbox} - 1$ _{Square (2, 6), Tealbox Patch}

$x_{31\tealbox} \ge u_{3\tealbox} + v_{1\tealbox} - 1$ _{Square (3, 1), Tealbox Patch}

$x_{32\tealbox} \ge u_{3\tealbox} + v_{2\tealbox} - 1$ _{Square (3, 2), Tealbox Patch}

$x_{33\tealbox} \ge u_{3\tealbox} + v_{3\tealbox} - 1$ _{Square (3, 3), Tealbox Patch}

$x_{34\tealbox} \ge u_{3\tealbox} + v_{4\tealbox} - 1$ _{Square (3, 4), Tealbox Patch}

$x_{35\tealbox} \ge u_{3\tealbox} + v_{5\tealbox} - 1$ _{Square (3, 5), Tealbox Patch}

$x_{36\tealbox} \ge u_{3\tealbox} + v_{6\tealbox} - 1$ _{Square (3, 6), Tealbox Patch}

$x_{41\tealbox} \ge u_{4\tealbox} + v_{1\tealbox} - 1$ _{Square (4, 1), Tealbox Patch}

$x_{42\tealbox} \ge u_{4\tealbox} + v_{2\tealbox} - 1$ _{Square (4, 2), Tealbox Patch}

$x_{43\tealbox} \ge u_{4\tealbox} + v_{3\tealbox} - 1$ _{Square (4, 3), Tealbox Patch}

$x_{44\tealbox} \ge u_{4\tealbox} + v_{4\tealbox} - 1$ _{Square (4, 4), Tealbox Patch}

$x_{45\tealbox} \ge u_{4\tealbox} + v_{5\tealbox} - 1$ _{Square (4, 5), Tealbox Patch}

$x_{46\tealbox} \ge u_{4\tealbox} + v_{6\tealbox} - 1$ _{Square (4, 6), Tealbox Patch}

$x_{51\tealbox} \ge u_{5\tealbox} + v_{1\tealbox} - 1$ _{Square (5, 1), Tealbox Patch}

$x_{52\tealbox} \ge u_{5\tealbox} + v_{2\tealbox} - 1$ _{Square (5, 2), Tealbox Patch}

$x_{53\tealbox} \ge u_{5\tealbox} + v_{3\tealbox} - 1$ _{Square (5, 3), Tealbox Patch}

$x_{54\tealbox} \ge u_{5\tealbox} + v_{4\tealbox} - 1$ _{Square (5, 4), Tealbox Patch}

$x_{55\tealbox} \ge u_{5\tealbox} + v_{5\tealbox} - 1$ _{Square (5, 5), Tealbox Patch}

$x_{56\tealbox} \ge u_{5\tealbox} + v_{6\tealbox} - 1$ _{Square (5, 6), Tealbox Patch}

$x_{61\tealbox} \ge u_{6\tealbox} + v_{1\tealbox} - 1$ _{Square (6, 1), Tealbox Patch}

$x_{62\tealbox} \ge u_{6\tealbox} + v_{2\tealbox} - 1$ _{Square (6, 2), Tealbox Patch}

$x_{63\tealbox} \ge u_{6\tealbox} + v_{3\tealbox} - 1$ _{Square (6, 3), Tealbox Patch}

$x_{64\tealbox} \ge u_{6\tealbox} + v_{4\tealbox} - 1$ _{Square (6, 4), Tealbox Patch}

$x_{65\tealbox} \ge u_{6\tealbox} + v_{5\tealbox} - 1$ _{Square (6, 5), Tealbox Patch}

$x_{66\tealbox} \ge u_{6\tealbox} + v_{6\tealbox} - 1$ _{Square (6, 6), Tealbox Patch}

$x_{11\purplebox} \ge u_{1\purplebox} + v_{1\purplebox} - 1$ _{Square (1, 1), Purple Patch}

$x_{12\purplebox} \ge u_{1\purplebox} + v_{2\purplebox} - 1$ _{Square (1, 2), Purple Patch}

$x_{13\purplebox} \ge u_{1\purplebox} + v_{3\purplebox} - 1$ _{Square (1, 3), Purple Patch}

$x_{14\purplebox} \ge u_{1\purplebox} + v_{4\purplebox} - 1$ _{Square (1, 4), Purple Patch}

$x_{15\purplebox} \ge u_{1\purplebox} + v_{5\purplebox} - 1$ _{Square (1, 5), Purple Patch}

$x_{16\purplebox} \ge u_{1\purplebox} + v_{6\purplebox} - 1$ _{Square (1, 6), Purple Patch}

$x_{21\purplebox} \ge u_{2\purplebox} + v_{1\purplebox} - 1$ _{Square (2, 1), Purple Patch}

$x_{22\purplebox} \ge u_{2\purplebox} + v_{2\purplebox} - 1$ _{Square (2, 2), Purple Patch}

$x_{23\purplebox} \ge u_{2\purplebox} + v_{3\purplebox} - 1$ _{Square (2, 3), Purple Patch}

$x_{24\purplebox} \ge u_{2\purplebox} + v_{4\purplebox} - 1$ _{Square (2, 4), Purple Patch}

$x_{25\purplebox} \ge u_{2\purplebox} + v_{5\purplebox} - 1$ _{Square (2, 5), Purple Patch}

$x_{26\purplebox} \ge u_{2\purplebox} + v_{6\purplebox} - 1$ _{Square (2, 6), Purple Patch}

$x_{31\purplebox} \ge u_{3\purplebox} + v_{1\purplebox} - 1$ _{Square (3, 1), Purple Patch}

$x_{32\purplebox} \ge u_{3\purplebox} + v_{2\purplebox} - 1$ _{Square (3, 2), Purple Patch}

$x_{33\purplebox} \ge u_{3\purplebox} + v_{3\purplebox} - 1$ _{Square (3, 3), Purple Patch}

$x_{34\purplebox} \ge u_{3\purplebox} + v_{4\purplebox} - 1$ _{Square (3, 4), Purple Patch}

$x_{35\purplebox} \ge u_{3\purplebox} + v_{5\purplebox} - 1$ _{Square (3, 5), Purple Patch}

$x_{36\purplebox} \ge u_{3\purplebox} + v_{6\purplebox} - 1$ _{Square (3, 6), Purple Patch}

$x_{41\purplebox} \ge u_{4\purplebox} + v_{1\purplebox} - 1$ _{Square (4, 1), Purple Patch}

$x_{42\purplebox} \ge u_{4\purplebox} + v_{2\purplebox} - 1$ _{Square (4, 2), Purple Patch}

$x_{43\purplebox} \ge u_{4\purplebox} + v_{3\purplebox} - 1$ _{Square (4, 3), Purple Patch}

$x_{44\purplebox} \ge u_{4\purplebox} + v_{4\purplebox} - 1$ _{Square (4, 4), Purple Patch}

$x_{45\purplebox} \ge u_{4\purplebox} + v_{5\purplebox} - 1$ _{Square (4, 5), Purple Patch}

$x_{46\purplebox} \ge u_{4\purplebox} + v_{6\purplebox} - 1$ _{Square (4, 6), Purple Patch}

$x_{51\purplebox} \ge u_{5\purplebox} + v_{1\purplebox} - 1$ _{Square (5, 1), Purple Patch}

$x_{52\purplebox} \ge u_{5\purplebox} + v_{2\purplebox} - 1$ _{Square (5, 2), Purple Patch}

$x_{53\purplebox} \ge u_{5\purplebox} + v_{3\purplebox} - 1$ _{Square (5, 3), Purple Patch}

$x_{54\purplebox} \ge u_{5\purplebox} + v_{4\purplebox} - 1$ _{Square (5, 4), Purple Patch}

$x_{55\purplebox} \ge u_{5\purplebox} + v_{5\purplebox} - 1$ _{Square (5, 5), Purple Patch}

$x_{56\purplebox} \ge u_{5\purplebox} + v_{6\purplebox} - 1$ _{Square (5, 6), Purple Patch}

$x_{61\purplebox} \ge u_{6\purplebox} + v_{1\purplebox} - 1$ _{Square (6, 1), Purple Patch}

$x_{62\purplebox} \ge u_{6\purplebox} + v_{2\purplebox} - 1$ _{Square (6, 2), Purple Patch}

$x_{63\purplebox} \ge u_{6\purplebox} + v_{3\purplebox} - 1$ _{Square (6, 3), Purple Patch}

$x_{64\purplebox} \ge u_{6\purplebox} + v_{4\purplebox} - 1$ _{Square (6, 4), Purple Patch}

$x_{65\purplebox} \ge u_{6\purplebox} + v_{5\purplebox} - 1$ _{Square (6, 5), Purple Patch}

$x_{66\purplebox} \ge u_{6\purplebox} + v_{6\purplebox} - 1$ _{Square (6, 6), Purple Patch}

$x_{11\greenbox} \ge u_{1\greenbox} + v_{1\greenbox} - 1$ _{Square (1, 1), Green Patch}

$x_{12\greenbox} \ge u_{1\greenbox} + v_{2\greenbox} - 1$ _{Square (1, 2), Green Patch}

$x_{13\greenbox} \ge u_{1\greenbox} + v_{3\greenbox} - 1$ _{Square (1, 3), Green Patch}

$x_{14\greenbox} \ge u_{1\greenbox} + v_{4\greenbox} - 1$ _{Square (1, 4), Green Patch}

$x_{15\greenbox} \ge u_{1\greenbox} + v_{5\greenbox} - 1$ _{Square (1, 5), Green Patch}

$x_{16\greenbox} \ge u_{1\greenbox} + v_{6\greenbox} - 1$ _{Square (1, 6), Green Patch}

$x_{21\greenbox} \ge u_{2\greenbox} + v_{1\greenbox} - 1$ _{Square (2, 1), Green Patch}

$x_{22\greenbox} \ge u_{2\greenbox} + v_{2\greenbox} - 1$ _{Square (2, 2), Green Patch}

$x_{23\greenbox} \ge u_{2\greenbox} + v_{3\greenbox} - 1$ _{Square (2, 3), Green Patch}

$x_{24\greenbox} \ge u_{2\greenbox} + v_{4\greenbox} - 1$ _{Square (2, 4), Green Patch}

$x_{25\greenbox} \ge u_{2\greenbox} + v_{5\greenbox} - 1$ _{Square (2, 5), Green Patch}

$x_{26\greenbox} \ge u_{2\greenbox} + v_{6\greenbox} - 1$ _{Square (2, 6), Green Patch}

$x_{31\greenbox} \ge u_{3\greenbox} + v_{1\greenbox} - 1$ _{Square (3, 1), Green Patch}

$x_{32\greenbox} \ge u_{3\greenbox} + v_{2\greenbox} - 1$ _{Square (3, 2), Green Patch}

$x_{33\greenbox} \ge u_{3\greenbox} + v_{3\greenbox} - 1$ _{Square (3, 3), Green Patch}

$x_{34\greenbox} \ge u_{3\greenbox} + v_{4\greenbox} - 1$ _{Square (3, 4), Green Patch}

$x_{35\greenbox} \ge u_{3\greenbox} + v_{5\greenbox} - 1$ _{Square (3, 5), Green Patch}

$x_{36\greenbox} \ge u_{3\greenbox} + v_{6\greenbox} - 1$ _{Square (3, 6), Green Patch}

$x_{41\greenbox} \ge u_{4\greenbox} + v_{1\greenbox} - 1$ _{Square (4, 1), Green Patch}

$x_{42\greenbox} \ge u_{4\greenbox} + v_{2\greenbox} - 1$ _{Square (4, 2), Green Patch}

$x_{43\greenbox} \ge u_{4\greenbox} + v_{3\greenbox} - 1$ _{Square (4, 3), Green Patch}

$x_{44\greenbox} \ge u_{4\greenbox} + v_{4\greenbox} - 1$ _{Square (4, 4), Green Patch}

$x_{45\greenbox} \ge u_{4\greenbox} + v_{5\greenbox} - 1$ _{Square (4, 5), Green Patch}

$x_{46\greenbox} \ge u_{4\greenbox} + v_{6\greenbox} - 1$ _{Square (4, 6), Green Patch}

$x_{51\greenbox} \ge u_{5\greenbox} + v_{1\greenbox} - 1$ _{Square (5, 1), Green Patch}

$x_{52\greenbox} \ge u_{5\greenbox} + v_{2\greenbox} - 1$ _{Square (5, 2), Green Patch}

$x_{53\greenbox} \ge u_{5\greenbox} + v_{3\greenbox} - 1$ _{Square (5, 3), Green Patch}

$x_{54\greenbox} \ge u_{5\greenbox} + v_{4\greenbox} - 1$ _{Square (5, 4), Green Patch}

$x_{55\greenbox} \ge u_{5\greenbox} + v_{5\greenbox} - 1$ _{Square (5, 5), Green Patch}

$x_{56\greenbox} \ge u_{5\greenbox} + v_{6\greenbox} - 1$ _{Square (5, 6), Green Patch}

$x_{61\greenbox} \ge u_{6\greenbox} + v_{1\greenbox} - 1$ _{Square (6, 1), Green Patch}

$x_{62\greenbox} \ge u_{6\greenbox} + v_{2\greenbox} - 1$ _{Square (6, 2), Green Patch}

$x_{63\greenbox} \ge u_{6\greenbox} + v_{3\greenbox} - 1$ _{Square (6, 3), Green Patch}

$x_{64\greenbox} \ge u_{6\greenbox} + v_{4\greenbox} - 1$ _{Square (6, 4), Green Patch}

$x_{65\greenbox} \ge u_{6\greenbox} + v_{5\greenbox} - 1$ _{Square (6, 5), Green Patch}

$x_{66\greenbox} \ge u_{6\greenbox} + v_{6\greenbox} - 1$ _{Square (6, 6), Green Patch}

$x_{11\orangebox} \ge u_{1\orangebox} + v_{1\orangebox} - 1$ _{Square (1, 1), Orange Patch}

$x_{12\orangebox} \ge u_{1\orangebox} + v_{2\orangebox} - 1$ _{Square (1, 2), Orange Patch}

$x_{13\orangebox} \ge u_{1\orangebox} + v_{3\orangebox} - 1$ _{Square (1, 3), Orange Patch}

$x_{14\orangebox} \ge u_{1\orangebox} + v_{4\orangebox} - 1$ _{Square (1, 4), Orange Patch}

$x_{15\orangebox} \ge u_{1\orangebox} + v_{5\orangebox} - 1$ _{Square (1, 5), Orange Patch}

$x_{16\orangebox} \ge u_{1\orangebox} + v_{6\orangebox} - 1$ _{Square (1, 6), Orange Patch}

$x_{21\orangebox} \ge u_{2\orangebox} + v_{1\orangebox} - 1$ _{Square (2, 1), Orange Patch}

$x_{22\orangebox} \ge u_{2\orangebox} + v_{2\orangebox} - 1$ _{Square (2, 2), Orange Patch}

$x_{23\orangebox} \ge u_{2\orangebox} + v_{3\orangebox} - 1$ _{Square (2, 3), Orange Patch}

$x_{24\orangebox} \ge u_{2\orangebox} + v_{4\orangebox} - 1$ _{Square (2, 4), Orange Patch}

$x_{25\orangebox} \ge u_{2\orangebox} + v_{5\orangebox} - 1$ _{Square (2, 5), Orange Patch}

$x_{26\orangebox} \ge u_{2\orangebox} + v_{6\orangebox} - 1$ _{Square (2, 6), Orange Patch}

$x_{31\orangebox} \ge u_{3\orangebox} + v_{1\orangebox} - 1$ _{Square (3, 1), Orange Patch}

$x_{32\orangebox} \ge u_{3\orangebox} + v_{2\orangebox} - 1$ _{Square (3, 2), Orange Patch}

$x_{33\orangebox} \ge u_{3\orangebox} + v_{3\orangebox} - 1$ _{Square (3, 3), Orange Patch}

$x_{34\orangebox} \ge u_{3\orangebox} + v_{4\orangebox} - 1$ _{Square (3, 4), Orange Patch}

$x_{35\orangebox} \ge u_{3\orangebox} + v_{5\orangebox} - 1$ _{Square (3, 5), Orange Patch}

$x_{36\orangebox} \ge u_{3\orangebox} + v_{6\orangebox} - 1$ _{Square (3, 6), Orange Patch}

$x_{41\orangebox} \ge u_{4\orangebox} + v_{1\orangebox} - 1$ _{Square (4, 1), Orange Patch}

$x_{42\orangebox} \ge u_{4\orangebox} + v_{2\orangebox} - 1$ _{Square (4, 2), Orange Patch}

$x_{43\orangebox} \ge u_{4\orangebox} + v_{3\orangebox} - 1$ _{Square (4, 3), Orange Patch}

$x_{44\orangebox} \ge u_{4\orangebox} + v_{4\orangebox} - 1$ _{Square (4, 4), Orange Patch}

$x_{45\orangebox} \ge u_{4\orangebox} + v_{5\orangebox} - 1$ _{Square (4, 5), Orange Patch}

$x_{46\orangebox} \ge u_{4\orangebox} + v_{6\orangebox} - 1$ _{Square (4, 6), Orange Patch}

$x_{51\orangebox} \ge u_{5\orangebox} + v_{1\orangebox} - 1$ _{Square (5, 1), Orange Patch}

$x_{52\orangebox} \ge u_{5\orangebox} + v_{2\orangebox} - 1$ _{Square (5, 2), Orange Patch}

$x_{53\orangebox} \ge u_{5\orangebox} + v_{3\orangebox} - 1$ _{Square (5, 3), Orange Patch}

$x_{54\orangebox} \ge u_{5\orangebox} + v_{4\orangebox} - 1$ _{Square (5, 4), Orange Patch}

$x_{55\orangebox} \ge u_{5\orangebox} + v_{5\orangebox} - 1$ _{Square (5, 5), Orange Patch}

$x_{56\orangebox} \ge u_{5\orangebox} + v_{6\orangebox} - 1$ _{Square (5, 6), Orange Patch}

$x_{61\orangebox} \ge u_{6\orangebox} + v_{1\orangebox} - 1$ _{Square (6, 1), Orange Patch}

$x_{62\orangebox} \ge u_{6\orangebox} + v_{2\orangebox} - 1$ _{Square (6, 2), Orange Patch}

$x_{63\orangebox} \ge u_{6\orangebox} + v_{3\orangebox} - 1$ _{Square (6, 3), Orange Patch}

$x_{64\orangebox} \ge u_{6\orangebox} + v_{4\orangebox} - 1$ _{Square (6, 4), Orange Patch}

$x_{65\orangebox} \ge u_{6\orangebox} + v_{5\orangebox} - 1$ _{Square (6, 5), Orange Patch}

$x_{66\orangebox} \ge u_{6\orangebox} + v_{6\orangebox} - 1$ _{Square (6, 6), Orange Patch}

$x_{11\redbox} \ge u_{1\redbox} + v_{1\redbox} - 1$ _{Square (1, 1), Red Patch}

$x_{12\redbox} \ge u_{1\redbox} + v_{2\redbox} - 1$ _{Square (1, 2), Red Patch}

$x_{13\redbox} \ge u_{1\redbox} + v_{3\redbox} - 1$ _{Square (1, 3), Red Patch}

$x_{14\redbox} \ge u_{1\redbox} + v_{4\redbox} - 1$ _{Square (1, 4), Red Patch}

$x_{15\redbox} \ge u_{1\redbox} + v_{5\redbox} - 1$ _{Square (1, 5), Red Patch}

$x_{16\redbox} \ge u_{1\redbox} + v_{6\redbox} - 1$ _{Square (1, 6), Red Patch}

$x_{21\redbox} \ge u_{2\redbox} + v_{1\redbox} - 1$ _{Square (2, 1), Red Patch}

$x_{22\redbox} \ge u_{2\redbox} + v_{2\redbox} - 1$ _{Square (2, 2), Red Patch}

$x_{23\redbox} \ge u_{2\redbox} + v_{3\redbox} - 1$ _{Square (2, 3), Red Patch}

$x_{24\redbox} \ge u_{2\redbox} + v_{4\redbox} - 1$ _{Square (2, 4), Red Patch}

$x_{25\redbox} \ge u_{2\redbox} + v_{5\redbox} - 1$ _{Square (2, 5), Red Patch}

$x_{26\redbox} \ge u_{2\redbox} + v_{6\redbox} - 1$ _{Square (2, 6), Red Patch}

$x_{31\redbox} \ge u_{3\redbox} + v_{1\redbox} - 1$ _{Square (3, 1), Red Patch}

$x_{32\redbox} \ge u_{3\redbox} + v_{2\redbox} - 1$ _{Square (3, 2), Red Patch}

$x_{33\redbox} \ge u_{3\redbox} + v_{3\redbox} - 1$ _{Square (3, 3), Red Patch}

$x_{34\redbox} \ge u_{3\redbox} + v_{4\redbox} - 1$ _{Square (3, 4), Red Patch}

$x_{35\redbox} \ge u_{3\redbox} + v_{5\redbox} - 1$ _{Square (3, 5), Red Patch}

$x_{36\redbox} \ge u_{3\redbox} + v_{6\redbox} - 1$ _{Square (3, 6), Red Patch}

$x_{41\redbox} \ge u_{4\redbox} + v_{1\redbox} - 1$ _{Square (4, 1), Red Patch}

$x_{42\redbox} \ge u_{4\redbox} + v_{2\redbox} - 1$ _{Square (4, 2), Red Patch}

$x_{43\redbox} \ge u_{4\redbox} + v_{3\redbox} - 1$ _{Square (4, 3), Red Patch}

$x_{44\redbox} \ge u_{4\redbox} + v_{4\redbox} - 1$ _{Square (4, 4), Red Patch}

$x_{45\redbox} \ge u_{4\redbox} + v_{5\redbox} - 1$ _{Square (4, 5), Red Patch}

$x_{46\redbox} \ge u_{4\redbox} + v_{6\redbox} - 1$ _{Square (4, 6), Red Patch}

$x_{51\redbox} \ge u_{5\redbox} + v_{1\redbox} - 1$ _{Square (5, 1), Red Patch}

$x_{52\redbox} \ge u_{5\redbox} + v_{2\redbox} - 1$ _{Square (5, 2), Red Patch}

$x_{53\redbox} \ge u_{5\redbox} + v_{3\redbox} - 1$ _{Square (5, 3), Red Patch}

$x_{54\redbox} \ge u_{5\redbox} + v_{4\redbox} - 1$ _{Square (5, 4), Red Patch}

$x_{55\redbox} \ge u_{5\redbox} + v_{5\redbox} - 1$ _{Square (5, 5), Red Patch}

$x_{56\redbox} \ge u_{5\redbox} + v_{6\redbox} - 1$ _{Square (5, 6), Red Patch}

$x_{61\redbox} \ge u_{6\redbox} + v_{1\redbox} - 1$ _{Square (6, 1), Red Patch}

$x_{62\redbox} \ge u_{6\redbox} + v_{2\redbox} - 1$ _{Square (6, 2), Red Patch}

$x_{63\redbox} \ge u_{6\redbox} + v_{3\redbox} - 1$ _{Square (6, 3), Red Patch}

$x_{64\redbox} \ge u_{6\redbox} + v_{4\redbox} - 1$ _{Square (6, 4), Red Patch}

$x_{65\redbox} \ge u_{6\redbox} + v_{5\redbox} - 1$ _{Square (6, 5), Red Patch}

$x_{66\redbox} \ge u_{6\redbox} + v_{6\redbox} - 1$ _{Square (6, 6), Red Patch}

$x_{11\bluebox} \ge u_{1\bluebox} + v_{1\bluebox} - 1$ _{Square (1, 1), Blue Patch}

$x_{12\bluebox} \ge u_{1\bluebox} + v_{2\bluebox} - 1$ _{Square (1, 2), Blue Patch}

$x_{13\bluebox} \ge u_{1\bluebox} + v_{3\bluebox} - 1$ _{Square (1, 3), Blue Patch}

$x_{14\bluebox} \ge u_{1\bluebox} + v_{4\bluebox} - 1$ _{Square (1, 4), Blue Patch}

$x_{15\bluebox} \ge u_{1\bluebox} + v_{5\bluebox} - 1$ _{Square (1, 5), Blue Patch}

$x_{16\bluebox} \ge u_{1\bluebox} + v_{6\bluebox} - 1$ _{Square (1, 6), Blue Patch}

$x_{21\bluebox} \ge u_{2\bluebox} + v_{1\bluebox} - 1$ _{Square (2, 1), Blue Patch}

$x_{22\bluebox} \ge u_{2\bluebox} + v_{2\bluebox} - 1$ _{Square (2, 2), Blue Patch}

$x_{23\bluebox} \ge u_{2\bluebox} + v_{3\bluebox} - 1$ _{Square (2, 3), Blue Patch}

$x_{24\bluebox} \ge u_{2\bluebox} + v_{4\bluebox} - 1$ _{Square (2, 4), Blue Patch}

$x_{25\bluebox} \ge u_{2\bluebox} + v_{5\bluebox} - 1$ _{Square (2, 5), Blue Patch}

$x_{26\bluebox} \ge u_{2\bluebox} + v_{6\bluebox} - 1$ _{Square (2, 6), Blue Patch}

$x_{31\bluebox} \ge u_{3\bluebox} + v_{1\bluebox} - 1$ _{Square (3, 1), Blue Patch}

$x_{32\bluebox} \ge u_{3\bluebox} + v_{2\bluebox} - 1$ _{Square (3, 2), Blue Patch}

$x_{33\bluebox} \ge u_{3\bluebox} + v_{3\bluebox} - 1$ _{Square (3, 3), Blue Patch}

$x_{34\bluebox} \ge u_{3\bluebox} + v_{4\bluebox} - 1$ _{Square (3, 4), Blue Patch}

$x_{35\bluebox} \ge u_{3\bluebox} + v_{5\bluebox} - 1$ _{Square (3, 5), Blue Patch}

$x_{36\bluebox} \ge u_{3\bluebox} + v_{6\bluebox} - 1$ _{Square (3, 6), Blue Patch}

$x_{41\bluebox} \ge u_{4\bluebox} + v_{1\bluebox} - 1$ _{Square (4, 1), Blue Patch}

$x_{42\bluebox} \ge u_{4\bluebox} + v_{2\bluebox} - 1$ _{Square (4, 2), Blue Patch}

$x_{43\bluebox} \ge u_{4\bluebox} + v_{3\bluebox} - 1$ _{Square (4, 3), Blue Patch}

$x_{44\bluebox} \ge u_{4\bluebox} + v_{4\bluebox} - 1$ _{Square (4, 4), Blue Patch}

$x_{45\bluebox} \ge u_{4\bluebox} + v_{5\bluebox} - 1$ _{Square (4, 5), Blue Patch}

$x_{46\bluebox} \ge u_{4\bluebox} + v_{6\bluebox} - 1$ _{Square (4, 6), Blue Patch}

$x_{51\bluebox} \ge u_{5\bluebox} + v_{1\bluebox} - 1$ _{Square (5, 1), Blue Patch}

$x_{52\bluebox} \ge u_{5\bluebox} + v_{2\bluebox} - 1$ _{Square (5, 2), Blue Patch}

$x_{53\bluebox} \ge u_{5\bluebox} + v_{3\bluebox} - 1$ _{Square (5, 3), Blue Patch}

$x_{54\bluebox} \ge u_{5\bluebox} + v_{4\bluebox} - 1$ _{Square (5, 4), Blue Patch}

$x_{55\bluebox} \ge u_{5\bluebox} + v_{5\bluebox} - 1$ _{Square (5, 5), Blue Patch}

$x_{56\bluebox} \ge u_{5\bluebox} + v_{6\bluebox} - 1$ _{Square (5, 6), Blue Patch}

$x_{61\bluebox} \ge u_{6\bluebox} + v_{1\bluebox} - 1$ _{Square (6, 1), Blue Patch}

$x_{62\bluebox} \ge u_{6\bluebox} + v_{2\bluebox} - 1$ _{Square (6, 2), Blue Patch}

$x_{63\bluebox} \ge u_{6\bluebox} + v_{3\bluebox} - 1$ _{Square (6, 3), Blue Patch}

$x_{64\bluebox} \ge u_{6\bluebox} + v_{4\bluebox} - 1$ _{Square (6, 4), Blue Patch}

$x_{65\bluebox} \ge u_{6\bluebox} + v_{5\bluebox} - 1$ _{Square (6, 5), Blue Patch}

$x_{66\bluebox} \ge u_{6\bluebox} + v_{6\bluebox} - 1$ _{Square (6, 6), Blue Patch}

$x_{11\magentabox} \ge u_{1\magentabox} + v_{1\magentabox} - 1$ _{Square (1, 1), Magenta Patch}

$x_{12\magentabox} \ge u_{1\magentabox} + v_{2\magentabox} - 1$ _{Square (1, 2), Magenta Patch}

$x_{13\magentabox} \ge u_{1\magentabox} + v_{3\magentabox} - 1$ _{Square (1, 3), Magenta Patch}

$x_{14\magentabox} \ge u_{1\magentabox} + v_{4\magentabox} - 1$ _{Square (1, 4), Magenta Patch}

$x_{15\magentabox} \ge u_{1\magentabox} + v_{5\magentabox} - 1$ _{Square (1, 5), Magenta Patch}

$x_{16\magentabox} \ge u_{1\magentabox} + v_{6\magentabox} - 1$ _{Square (1, 6), Magenta Patch}

$x_{21\magentabox} \ge u_{2\magentabox} + v_{1\magentabox} - 1$ _{Square (2, 1), Magenta Patch}

$x_{22\magentabox} \ge u_{2\magentabox} + v_{2\magentabox} - 1$ _{Square (2, 2), Magenta Patch}

$x_{23\magentabox} \ge u_{2\magentabox} + v_{3\magentabox} - 1$ _{Square (2, 3), Magenta Patch}

$x_{24\magentabox} \ge u_{2\magentabox} + v_{4\magentabox} - 1$ _{Square (2, 4), Magenta Patch}

$x_{25\magentabox} \ge u_{2\magentabox} + v_{5\magentabox} - 1$ _{Square (2, 5), Magenta Patch}

$x_{26\magentabox} \ge u_{2\magentabox} + v_{6\magentabox} - 1$ _{Square (2, 6), Magenta Patch}

$x_{31\magentabox} \ge u_{3\magentabox} + v_{1\magentabox} - 1$ _{Square (3, 1), Magenta Patch}

$x_{32\magentabox} \ge u_{3\magentabox} + v_{2\magentabox} - 1$ _{Square (3, 2), Magenta Patch}

$x_{33\magentabox} \ge u_{3\magentabox} + v_{3\magentabox} - 1$ _{Square (3, 3), Magenta Patch}

$x_{34\magentabox} \ge u_{3\magentabox} + v_{4\magentabox} - 1$ _{Square (3, 4), Magenta Patch}

$x_{35\magentabox} \ge u_{3\magentabox} + v_{5\magentabox} - 1$ _{Square (3, 5), Magenta Patch}

$x_{36\magentabox} \ge u_{3\magentabox} + v_{6\magentabox} - 1$ _{Square (3, 6), Magenta Patch}

$x_{41\magentabox} \ge u_{4\magentabox} + v_{1\magentabox} - 1$ _{Square (4, 1), Magenta Patch}

$x_{42\magentabox} \ge u_{4\magentabox} + v_{2\magentabox} - 1$ _{Square (4, 2), Magenta Patch}

$x_{43\magentabox} \ge u_{4\magentabox} + v_{3\magentabox} - 1$ _{Square (4, 3), Magenta Patch}

$x_{44\magentabox} \ge u_{4\magentabox} + v_{4\magentabox} - 1$ _{Square (4, 4), Magenta Patch}

$x_{45\magentabox} \ge u_{4\magentabox} + v_{5\magentabox} - 1$ _{Square (4, 5), Magenta Patch}

$x_{46\magentabox} \ge u_{4\magentabox} + v_{6\magentabox} - 1$ _{Square (4, 6), Magenta Patch}

$x_{51\magentabox} \ge u_{5\magentabox} + v_{1\magentabox} - 1$ _{Square (5, 1), Magenta Patch}

$x_{52\magentabox} \ge u_{5\magentabox} + v_{2\magentabox} - 1$ _{Square (5, 2), Magenta Patch}

$x_{53\magentabox} \ge u_{5\magentabox} + v_{3\magentabox} - 1$ _{Square (5, 3), Magenta Patch}

$x_{54\magentabox} \ge u_{5\magentabox} + v_{4\magentabox} - 1$ _{Square (5, 4), Magenta Patch}

$x_{55\magentabox} \ge u_{5\magentabox} + v_{5\magentabox} - 1$ _{Square (5, 5), Magenta Patch}

$x_{56\magentabox} \ge u_{5\magentabox} + v_{6\magentabox} - 1$ _{Square (5, 6), Magenta Patch}

$x_{61\magentabox} \ge u_{6\magentabox} + v_{1\magentabox} - 1$ _{Square (6, 1), Magenta Patch}

$x_{62\magentabox} \ge u_{6\magentabox} + v_{2\magentabox} - 1$ _{Square (6, 2), Magenta Patch}

$x_{63\magentabox} \ge u_{6\magentabox} + v_{3\magentabox} - 1$ _{Square (6, 3), Magenta Patch}

$x_{64\magentabox} \ge u_{6\magentabox} + v_{4\magentabox} - 1$ _{Square (6, 4), Magenta Patch}

$x_{65\magentabox} \ge u_{6\magentabox} + v_{5\magentabox} - 1$ _{Square (6, 5), Magenta Patch}

$x_{66\magentabox} \ge u_{6\magentabox} + v_{6\magentabox} - 1$ _{Square (6, 6), Magenta Patch}

$x_{11\brickbox} \ge u_{1\brickbox} + v_{1\brickbox} - 1$ _{Square (1, 1), Brick Patch}

$x_{12\brickbox} \ge u_{1\brickbox} + v_{2\brickbox} - 1$ _{Square (1, 2), Brick Patch}

$x_{13\brickbox} \ge u_{1\brickbox} + v_{3\brickbox} - 1$ _{Square (1, 3), Brick Patch}

$x_{14\brickbox} \ge u_{1\brickbox} + v_{4\brickbox} - 1$ _{Square (1, 4), Brick Patch}

$x_{15\brickbox} \ge u_{1\brickbox} + v_{5\brickbox} - 1$ _{Square (1, 5), Brick Patch}

$x_{16\brickbox} \ge u_{1\brickbox} + v_{6\brickbox} - 1$ _{Square (1, 6), Brick Patch}

$x_{21\brickbox} \ge u_{2\brickbox} + v_{1\brickbox} - 1$ _{Square (2, 1), Brick Patch}

$x_{22\brickbox} \ge u_{2\brickbox} + v_{2\brickbox} - 1$ _{Square (2, 2), Brick Patch}

$x_{23\brickbox} \ge u_{2\brickbox} + v_{3\brickbox} - 1$ _{Square (2, 3), Brick Patch}

$x_{24\brickbox} \ge u_{2\brickbox} + v_{4\brickbox} - 1$ _{Square (2, 4), Brick Patch}

$x_{25\brickbox} \ge u_{2\brickbox} + v_{5\brickbox} - 1$ _{Square (2, 5), Brick Patch}

$x_{26\brickbox} \ge u_{2\brickbox} + v_{6\brickbox} - 1$ _{Square (2, 6), Brick Patch}

$x_{31\brickbox} \ge u_{3\brickbox} + v_{1\brickbox} - 1$ _{Square (3, 1), Brick Patch}

$x_{32\brickbox} \ge u_{3\brickbox} + v_{2\brickbox} - 1$ _{Square (3, 2), Brick Patch}

$x_{33\brickbox} \ge u_{3\brickbox} + v_{3\brickbox} - 1$ _{Square (3, 3), Brick Patch}

$x_{34\brickbox} \ge u_{3\brickbox} + v_{4\brickbox} - 1$ _{Square (3, 4), Brick Patch}

$x_{35\brickbox} \ge u_{3\brickbox} + v_{5\brickbox} - 1$ _{Square (3, 5), Brick Patch}

$x_{36\brickbox} \ge u_{3\brickbox} + v_{6\brickbox} - 1$ _{Square (3, 6), Brick Patch}

$x_{41\brickbox} \ge u_{4\brickbox} + v_{1\brickbox} - 1$ _{Square (4, 1), Brick Patch}

$x_{42\brickbox} \ge u_{4\brickbox} + v_{2\brickbox} - 1$ _{Square (4, 2), Brick Patch}

$x_{43\brickbox} \ge u_{4\brickbox} + v_{3\brickbox} - 1$ _{Square (4, 3), Brick Patch}

$x_{44\brickbox} \ge u_{4\brickbox} + v_{4\brickbox} - 1$ _{Square (4, 4), Brick Patch}

$x_{45\brickbox} \ge u_{4\brickbox} + v_{5\brickbox} - 1$ _{Square (4, 5), Brick Patch}

$x_{46\brickbox} \ge u_{4\brickbox} + v_{6\brickbox} - 1$ _{Square (4, 6), Brick Patch}

$x_{51\brickbox} \ge u_{5\brickbox} + v_{1\brickbox} - 1$ _{Square (5, 1), Brick Patch}

$x_{52\brickbox} \ge u_{5\brickbox} + v_{2\brickbox} - 1$ _{Square (5, 2), Brick Patch}

$x_{53\brickbox} \ge u_{5\brickbox} + v_{3\brickbox} - 1$ _{Square (5, 3), Brick Patch}

$x_{54\brickbox} \ge u_{5\brickbox} + v_{4\brickbox} - 1$ _{Square (5, 4), Brick Patch}

$x_{55\brickbox} \ge u_{5\brickbox} + v_{5\brickbox} - 1$ _{Square (5, 5), Brick Patch}

$x_{56\brickbox} \ge u_{5\brickbox} + v_{6\brickbox} - 1$ _{Square (5, 6), Brick Patch}

$x_{61\brickbox} \ge u_{6\brickbox} + v_{1\brickbox} - 1$ _{Square (6, 1), Brick Patch}

$x_{62\brickbox} \ge u_{6\brickbox} + v_{2\brickbox} - 1$ _{Square (6, 2), Brick Patch}

$x_{63\brickbox} \ge u_{6\brickbox} + v_{3\brickbox} - 1$ _{Square (6, 3), Brick Patch}

$x_{64\brickbox} \ge u_{6\brickbox} + v_{4\brickbox} - 1$ _{Square (6, 4), Brick Patch}

$x_{65\brickbox} \ge u_{6\brickbox} + v_{5\brickbox} - 1$ _{Square (6, 5), Brick Patch}

$x_{66\brickbox} \ge u_{6\brickbox} + v_{6\brickbox} - 1$ _{Square (6, 6), Brick Patch}

$x_{11\brownbox} \ge u_{1\brownbox} + v_{1\brownbox} - 1$ _{Square (1, 1), Brown Patch}

$x_{12\brownbox} \ge u_{1\brownbox} + v_{2\brownbox} - 1$ _{Square (1, 2), Brown Patch}

$x_{13\brownbox} \ge u_{1\brownbox} + v_{3\brownbox} - 1$ _{Square (1, 3), Brown Patch}

$x_{14\brownbox} \ge u_{1\brownbox} + v_{4\brownbox} - 1$ _{Square (1, 4), Brown Patch}

$x_{15\brownbox} \ge u_{1\brownbox} + v_{5\brownbox} - 1$ _{Square (1, 5), Brown Patch}

$x_{16\brownbox} \ge u_{1\brownbox} + v_{6\brownbox} - 1$ _{Square (1, 6), Brown Patch}

$x_{21\brownbox} \ge u_{2\brownbox} + v_{1\brownbox} - 1$ _{Square (2, 1), Brown Patch}

$x_{22\brownbox} \ge u_{2\brownbox} + v_{2\brownbox} - 1$ _{Square (2, 2), Brown Patch}

$x_{23\brownbox} \ge u_{2\brownbox} + v_{3\brownbox} - 1$ _{Square (2, 3), Brown Patch}

$x_{24\brownbox} \ge u_{2\brownbox} + v_{4\brownbox} - 1$ _{Square (2, 4), Brown Patch}

$x_{25\brownbox} \ge u_{2\brownbox} + v_{5\brownbox} - 1$ _{Square (2, 5), Brown Patch}

$x_{26\brownbox} \ge u_{2\brownbox} + v_{6\brownbox} - 1$ _{Square (2, 6), Brown Patch}

$x_{31\brownbox} \ge u_{3\brownbox} + v_{1\brownbox} - 1$ _{Square (3, 1), Brown Patch}

$x_{32\brownbox} \ge u_{3\brownbox} + v_{2\brownbox} - 1$ _{Square (3, 2), Brown Patch}

$x_{33\brownbox} \ge u_{3\brownbox} + v_{3\brownbox} - 1$ _{Square (3, 3), Brown Patch}

$x_{34\brownbox} \ge u_{3\brownbox} + v_{4\brownbox} - 1$ _{Square (3, 4), Brown Patch}

$x_{35\brownbox} \ge u_{3\brownbox} + v_{5\brownbox} - 1$ _{Square (3, 5), Brown Patch}

$x_{36\brownbox} \ge u_{3\brownbox} + v_{6\brownbox} - 1$ _{Square (3, 6), Brown Patch}

$x_{41\brownbox} \ge u_{4\brownbox} + v_{1\brownbox} - 1$ _{Square (4, 1), Brown Patch}

$x_{42\brownbox} \ge u_{4\brownbox} + v_{2\brownbox} - 1$ _{Square (4, 2), Brown Patch}

$x_{43\brownbox} \ge u_{4\brownbox} + v_{3\brownbox} - 1$ _{Square (4, 3), Brown Patch}

$x_{44\brownbox} \ge u_{4\brownbox} + v_{4\brownbox} - 1$ _{Square (4, 4), Brown Patch}

$x_{45\brownbox} \ge u_{4\brownbox} + v_{5\brownbox} - 1$ _{Square (4, 5), Brown Patch}

$x_{46\brownbox} \ge u_{4\brownbox} + v_{6\brownbox} - 1$ _{Square (4, 6), Brown Patch}

$x_{51\brownbox} \ge u_{5\brownbox} + v_{1\brownbox} - 1$ _{Square (5, 1), Brown Patch}

$x_{52\brownbox} \ge u_{5\brownbox} + v_{2\brownbox} - 1$ _{Square (5, 2), Brown Patch}

$x_{53\brownbox} \ge u_{5\brownbox} + v_{3\brownbox} - 1$ _{Square (5, 3), Brown Patch}

$x_{54\brownbox} \ge u_{5\brownbox} + v_{4\brownbox} - 1$ _{Square (5, 4), Brown Patch}

$x_{55\brownbox} \ge u_{5\brownbox} + v_{5\brownbox} - 1$ _{Square (5, 5), Brown Patch}

$x_{56\brownbox} \ge u_{5\brownbox} + v_{6\brownbox} - 1$ _{Square (5, 6), Brown Patch}

$x_{61\brownbox} \ge u_{6\brownbox} + v_{1\brownbox} - 1$ _{Square (6, 1), Brown Patch}

$x_{62\brownbox} \ge u_{6\brownbox} + v_{2\brownbox} - 1$ _{Square (6, 2), Brown Patch}

$x_{63\brownbox} \ge u_{6\brownbox} + v_{3\brownbox} - 1$ _{Square (6, 3), Brown Patch}

$x_{64\brownbox} \ge u_{6\brownbox} + v_{4\brownbox} - 1$ _{Square (6, 4), Brown Patch}

$x_{65\brownbox} \ge u_{6\brownbox} + v_{5\brownbox} - 1$ _{Square (6, 5), Brown Patch}

$x_{66\brownbox} \ge u_{6\brownbox} + v_{6\brownbox} - 1$ _{Square (6, 6), Brown Patch}

Seed Square Coverage Constraints

$x_{12\yellowbox} = 1$ _{Yellow Square}

$x_{14\tealbox} = 1$ _{Teal Square}

$x_{26\purplebox} = 1$ _{Purple Square}

$x_{31\tealbox} = 1$ _{Green Square}

$x_{33\orangebox} = 1$ _{Orange Square}

$x_{44\redbox} = 1$ _{Red Square}

$x_{46\bluebox} = 1$ _{Blue Square}

$x_{51\magentabox} = 1$ _{Magenta Square}

$x_{63\brickbox} = 1$ _{Brick Square}

$x_{65\brownbox} = 1$ _{Brown Square}

Required Area Constraints

$x_{11\yellowbox} + x_{12\yellowbox} + x_{13\yellowbox} + x_{14\yellowbox} + x_{15\yellowbox} + x_{16\yellowbox} + x_{21\yellowbox} + x_{22\yellowbox} + x_{23\yellowbox} + x_{24\yellowbox} + x_{25\yellowbox} + x_{26\yellowbox} + x_{31\yellowbox} + x_{32\yellowbox} + x_{33\yellowbox} + x_{34\yellowbox} + x_{35\yellowbox} + x_{36\yellowbox} + x_{41\yellowbox} + x_{42\yellowbox} + x_{43\yellowbox} + x_{44\yellowbox} + x_{45\yellowbox} + x_{46\yellowbox} + x_{51\yellowbox} + x_{52\yellowbox} + x_{53\yellowbox} + x_{54\yellowbox} + x_{55\yellowbox} + x_{56\yellowbox} + x_{61\yellowbox} + x_{62\yellowbox} + x_{63\yellowbox} + x_{64\yellowbox} + x_{65\yellowbox} + x_{66\yellowbox} = 2$ _{Yellow Patch}

$x_{11\tealbox} + x_{12\tealbox} + x_{13\tealbox} + x_{14\tealbox} + x_{15\tealbox} + x_{16\tealbox} + x_{21\tealbox} + x_{22\tealbox} + x_{23\tealbox} + x_{24\tealbox} + x_{25\tealbox}+ x_{26\tealbox} + x_{31\tealbox} + x_{32\tealbox} + x_{33\tealbox} + x_{34\tealbox} + x_{35\tealbox} + x_{36\tealbox} + x_{41\tealbox} + x_{42\tealbox} + x_{43\tealbox} + x_{44\tealbox} +x_{45\tealbox} + x_{46\tealbox} + x_{51\tealbox} + x_{52\tealbox} + x_{53\tealbox} + x_{54\tealbox} + x_{55\tealbox} + x_{56\tealbox} + x_{61\tealbox} + x_{62\tealbox} + x_{63\tealbox} + x_{64\tealbox} + x_{65\tealbox} + x_{66\tealbox} = 6$ _{Teal Patch}

$x_{11\purplebox} + x_{12\purplebox} + x_{13\purplebox} + x_{14\purplebox} + x_{15\purplebox} + x_{16\purplebox} + x_{21\purplebox} + x_{22\purplebox} + x_{23\purplebox} + x_{24\purplebox} + x_{25\purplebox} + x_{26\purplebox} + x_{31\purplebox} + x_{32\purplebox} + x_{33\purplebox} + x_{34\purplebox} + x_{35\purplebox} + x_{36\purplebox} + x_{41\purplebox} + x_{42\purplebox} + x_{43\purplebox} + x_{44\purplebox} + x_{45\purplebox} + x_{46\purplebox} + x_{51\purplebox} + x_{52\purplebox} + x_{53\purplebox} + x_{54\purplebox} + x_{55\purplebox} + x_{56\purplebox} + x_{61\purplebox} + x_{62\purplebox} + x_{63\purplebox} + x_{64\purplebox} + x_{65\purplebox} + x_{66\purplebox} = 2$ _{Purple Patch}

$x_{11\greenbox} + x_{12\greenbox} + x_{13\greenbox} + x_{14\greenbox} + x_{15\greenbox} + x_{16\greenbox} + x_{21\greenbox} + x_{22\greenbox} + x_{23\greenbox} + x_{24\greenbox} + x_{25\greenbox} + x_{26\greenbox} + x_{31\greenbox} + x_{32\greenbox} + x_{33\greenbox} + x_{34\greenbox} + x_{35\greenbox} + x_{36\greenbox} + x_{41\greenbox} + x_{42\greenbox} + x_{43\greenbox} + x_{44\greenbox} + x_{45\greenbox} + x_{46\greenbox} + x_{51\greenbox} + x_{52\greenbox} + x_{53\greenbox} + x_{54\greenbox} + x_{55\greenbox} + x_{56\greenbox} + x_{61\greenbox} + x_{62\greenbox} + x_{63\greenbox} + x_{64\greenbox} + x_{65\greenbox} + x_{66\greenbox} = 6$ _{Green Patch}

$x_{11\orangebox} + x_{12\orangebox} + x_{13\orangebox} + x_{14\orangebox} + x_{15\orangebox} + x_{16\orangebox} + x_{21\orangebox} + x_{22\orangebox} + x_{23\orangebox} + x_{24\orangebox} + x_{25\orangebox} + x_{26\orangebox} + x_{31\orangebox} + x_{32\orangebox} + x_{33\orangebox} + x_{34\orangebox} + x_{35\orangebox} + x_{36\orangebox} + x_{41\orangebox} + x_{42\orangebox} + x_{43\orangebox} + x_{44\orangebox} + x_{45\orangebox} + x_{46\orangebox} + x_{51\orangebox} + x_{52\orangebox} + x_{53\orangebox} + x_{54\orangebox} + x_{55\orangebox} + x_{56\orangebox} + x_{61\orangebox} + x_{62\orangebox} + x_{63\orangebox} + x_{64\orangebox} + x_{65\orangebox} + x_{66\orangebox} = 2$ _{Orange Patch}

$x_{11\redbox} + x_{12\redbox} + x_{13\redbox} + x_{14\redbox} + x_{15\redbox} + x_{16\redbox} + x_{21\redbox} + x_{22\redbox} + x_{23\redbox} + x_{24\redbox} + x_{25\redbox} + x_{26\redbox} + x_{31\redbox} + x_{32\redbox} + x_{33\redbox} + x_{34\redbox} + x_{35\redbox} + x_{36\redbox} + x_{41\redbox} + x_{42\redbox} + x_{43\redbox} + x_{44\redbox} + x_{45\redbox} + x_{46\redbox} + x_{51\redbox} + x_{52\redbox} + x_{53\redbox} + x_{54\redbox} + x_{55\redbox} + x_{56\redbox} + x_{61\redbox} + x_{62\redbox} + x_{63\redbox} + x_{64\redbox} + x_{65\redbox} + x_{66\redbox} = 4$ _{Red Patch}

$x_{11\bluebox} + x_{12\bluebox} + x_{13\bluebox} + x_{14\bluebox} + x_{15\bluebox} + x_{16\bluebox} + x_{21\bluebox} + x_{22\bluebox} + x_{23\bluebox} + x_{24\bluebox} + x_{25\bluebox}+ x_{26\bluebox} + x_{31\bluebox} + x_{32\bluebox} + x_{33\bluebox} + x_{34\bluebox} + x_{35\bluebox} + x_{36\bluebox} + x_{41\bluebox} + x_{42\bluebox} + x_{43\bluebox} + x_{44\bluebox} +x_{45\bluebox} + x_{46\bluebox} + x_{51\bluebox} + x_{52\bluebox} + x_{53\bluebox} + x_{54\bluebox} + x_{55\bluebox} + x_{56\bluebox} + x_{61\bluebox} + x_{62\bluebox} + x_{63\bluebox} + x_{64\bluebox} + x_{65\bluebox} + x_{66\bluebox} = 2$ _{Blue Patch}

$x_{11\magentabox} + x_{12\magentabox} + x_{13\magentabox} + x_{14\magentabox} + x_{15\magentabox} + x_{16\magentabox} + x_{21\magentabox} + x_{22\magentabox} + x_{23\magentabox} + x_{24\magentabox} + x_{25\magentabox} + x_{26\magentabox} + x_{31\magentabox} + x_{32\magentabox} + x_{33\magentabox} + x_{34\magentabox} + x_{35\magentabox} + x_{36\magentabox} + x_{41\magentabox} + x_{42\magentabox} + x_{43\magentabox} + x_{44\magentabox} + x_{45\magentabox} + x_{46\magentabox} + x_{51\magentabox} + x_{52\magentabox} + x_{53\magentabox} + x_{54\magentabox} + x_{55\magentabox} + x_{56\magentabox} + x_{61\magentabox} + x_{62\magentabox} + x_{63\magentabox} + x_{64\magentabox} + x_{65\magentabox} + x_{66\magentabox} = 2$ _{Magenta Patch}

$x_{11\brickbox} + x_{12\brickbox} + x_{13\brickbox} + x_{14\brickbox} + x_{15\brickbox} + x_{16\brickbox} + x_{21\brickbox} + x_{22\brickbox} + x_{23\brickbox} + x_{24\brickbox} + x_{25\brickbox} + x_{26\brickbox} + x_{31\brickbox} + x_{32\brickbox} + x_{33\brickbox} + x_{34\brickbox} + x_{35\brickbox} + x_{36\brickbox} + x_{41\brickbox} + x_{42\brickbox} + x_{43\brickbox} + x_{44\brickbox} + x_{45\brickbox} + x_{46\brickbox} + x_{51\brickbox} + x_{52\brickbox} + x_{53\brickbox} + x_{54\brickbox} + x_{55\brickbox} + x_{56\brickbox} + x_{61\brickbox} + x_{62\brickbox} + x_{63\brickbox} + x_{64\brickbox} + x_{65\brickbox} + x_{66\brickbox} = 6$ _{Brick Patch}

$x_{11\brownbox} + x_{12\brownbox} + x_{13\brownbox} + x_{14\brownbox} + x_{15\brownbox} + x_{16\brownbox} + x_{21\brownbox} + x_{22\brownbox} + x_{23\brownbox} + x_{24\brownbox} + x_{25\brownbox} + x_{26\brownbox} + x_{31\brownbox} + x_{32\brownbox} + x_{33\brownbox} + x_{34\brownbox} + x_{35\brownbox} + x_{36\brownbox} + x_{41\brownbox} + x_{42\brownbox} + x_{43\brownbox} + x_{44\brownbox} + x_{45\brownbox} + x_{46\brownbox} + x_{51\brownbox} + x_{52\brownbox} + x_{53\brownbox} + x_{54\brownbox} + x_{55\brownbox} + x_{56\brownbox} + x_{61\brownbox} + x_{62\brownbox} + x_{63\brownbox} + x_{64\brownbox} + x_{65\brownbox} + x_{66\brownbox} = 4$ _{Brown Patch}

Vertical-Patch Constraints

$w_{\orangebox} < h_{\orangebox}$ _{Orange Patch}

Square-Patch Constraints

$w_{\redbox} = h_{\redbox}$ _{Red Patch}

Integrity Constraints

$l_{\yellowbox} \in \N$ _{Yellow Patch}

$l_{\tealbox} \in \N$ _{Teal Patch}

$l_{\purplebox} \in \N$ _{Purple Patch}

$l_{\greenbox} \in \N$ _{Green Patch}

$l_{\orangebox} \in \N$ _{Orange Patch}

$l_{\redbox} \in \N$ _{Red Patch}

$l_{\bluebox} \in \N$ _{Blue Patch}

$l_{\magentabox} \in \N$ _{Magenta Patch}

$l_{\brickbox} \in \N$ _{Brick Patch}

$l_{\brownbox} \in \N$ _{Brown Patch}

$w_{\yellowbox} \in \N$ _{Yellow Patch}

$w_{\tealbox} \in \N$ _{Teal Patch}

$w_{\purplebox} \in \N$ _{Purple Patch}

$w_{\greenbox} \in \N$ _{Green Patch}

$w_{\orangebox} \in \N$ _{Orange Patch}

$w_{\redbox} \in \N$ _{Red Patch}

$w_{\bluebox} \in \N$ _{Blue Patch}

$w_{\magentabox} \in \N$ _{Magenta Patch}

$w_{\brickbox} \in \N$ _{Brick Patch}

$w_{\brownbox} \in \N$ _{Brown Patch}

$t_{\yellowbox} \in \N$ _{Yellow Patch}

$t_{\tealbox} \in \N$ _{Teal Patch}

$t_{\purplebox} \in \N$ _{Purple Patch}

$t_{\greenbox} \in \N$ _{Green Patch}

$t_{\orangebox} \in \N$ _{Orange Patch}

$t_{\redbox} \in \N$ _{Red Patch}

$t_{\bluebox} \in \N$ _{Blue Patch}

$t_{\magentabox} \in \N$ _{Magenta Patch}

$t_{\brickbox} \in \N$ _{Brick Patch}

$t_{\brownbox} \in \N$ _{Brown Patch}

$h_{\yellowbox} \in \N$ _{Yellow Patch}

$h_{\tealbox} \in \N$ _{Teal Patch}

$h_{\purplebox} \in \N$ _{Purple Patch}

$h_{\greenbox} \in \N$ _{Green Patch}

$h_{\orangebox} \in \N$ _{Orange Patch}

$h_{\redbox} \in \N$ _{Red Patch}

$h_{\bluebox} \in \N$ _{Blue Patch}

$h_{\magentabox} \in \N$ _{Magenta Patch}

$h_{\brickbox} \in \N$ _{Brick Patch}

$h_{\brownbox} \in \N$ _{Brown Patch}

Binarity Constraints

$u_{1\yellowbox} \in \B$ _{Row 1, Yellow Patch}

$u_{2\yellowbox} \in \B$ _{Row 2, Yellow Patch}

$u_{3\yellowbox} \in \B$ _{Row 3, Yellow Patch}

$u_{4\yellowbox} \in \B$ _{Row 4, Yellow Patch}

$u_{5\yellowbox} \in \B$ _{Row 5, Yellow Patch}

$u_{6\yellowbox} \in \B$ _{Row 6, Yellow Patch}

$u_{1\tealbox} \in \B$ _{Row 1, Tealbox Patch}

$u_{2\tealbox} \in \B$ _{Row 2, Tealbox Patch}

$u_{3\tealbox} \in \B$ _{Row 3, Tealbox Patch}

$u_{4\tealbox} \in \B$ _{Row 4, Tealbox Patch}

$u_{5\tealbox} \in \B$ _{Row 5, Tealbox Patch}

$u_{6\tealbox} \in \B$ _{Row 6, Tealbox Patch}

$u_{1\purplebox} \in \B$ _{Row 1, Purple Patch}

$u_{2\purplebox} \in \B$ _{Row 2, Purple Patch}

$u_{3\purplebox} \in \B$ _{Row 3, Purple Patch}

$u_{4\purplebox} \in \B$ _{Row 4, Purple Patch}

$u_{5\purplebox} \in \B$ _{Row 5, Purple Patch}

$u_{6\purplebox} \in \B$ _{Row 6, Purple Patch}

$u_{1\greenbox} \in \B$ _{Row 1, Green Patch}

$u_{2\greenbox} \in \B$ _{Row 2, Green Patch}

$u_{3\greenbox} \in \B$ _{Row 3, Green Patch}

$u_{4\greenbox} \in \B$ _{Row 4, Green Patch}

$u_{5\greenbox} \in \B$ _{Row 5, Green Patch}

$u_{6\greenbox} \in \B$ _{Row 6, Green Patch}

$u_{1\orangebox} \in \B$ _{Row 1, Orange Patch}

$u_{2\orangebox} \in \B$ _{Row 2, Orange Patch}

$u_{3\orangebox} \in \B$ _{Row 3, Orange Patch}

$u_{4\orangebox} \in \B$ _{Row 4, Orange Patch}

$u_{5\orangebox} \in \B$ _{Row 5, Orange Patch}

$u_{6\orangebox} \in \B$ _{Row 6, Orange Patch}

$u_{1\redbox} \in \B$ _{Row 1, Red Patch}

$u_{2\redbox} \in \B$ _{Row 2, Red Patch}

$u_{3\redbox} \in \B$ _{Row 3, Red Patch}

$u_{4\redbox} \in \B$ _{Row 4, Red Patch}

$u_{5\redbox} \in \B$ _{Row 5, Red Patch}

$u_{6\redbox} \in \B$ _{Row 6, Red Patch}

$u_{1\bluebox} \in \B$ _{Row 1, Blue Patch}

$u_{2\bluebox} \in \B$ _{Row 2, Blue Patch}

$u_{3\bluebox} \in \B$ _{Row 3, Blue Patch}

$u_{4\bluebox} \in \B$ _{Row 4, Blue Patch}

$u_{5\bluebox} \in \B$ _{Row 5, Blue Patch}

$u_{6\bluebox} \in \B$ _{Row 6, Blue Patch}

$u_{1\magentabox} \in \B$ _{Row 1, Magenta Patch}

$u_{2\magentabox} \in \B$ _{Row 2, Magenta Patch}

$u_{3\magentabox} \in \B$ _{Row 3, Magenta Patch}

$u_{4\magentabox} \in \B$ _{Row 4, Magenta Patch}

$u_{5\magentabox} \in \B$ _{Row 5, Magenta Patch}

$u_{6\magentabox} \in \B$ _{Row 6, Magenta Patch}

$u_{1\brickbox} \in \B$ _{Row 1, Brick Patch}

$u_{2\brickbox} \in \B$ _{Row 2, Brick Patch}

$u_{3\brickbox} \in \B$ _{Row 3, Brick Patch}

$u_{4\brickbox} \in \B$ _{Row 4, Brick Patch}

$u_{5\brickbox} \in \B$ _{Row 5, Brick Patch}

$u_{6\brickbox} \in \B$ _{Row 6, Brick Patch}

$u_{1\brownbox} \in \B$ _{Row 1, Brown Patch}

$u_{2\brownbox} \in \B$ _{Row 2, Brown Patch}

$u_{3\brownbox} \in \B$ _{Row 3, Brown Patch}

$u_{4\brownbox} \in \B$ _{Row 4, Brown Patch}

$u_{5\brownbox} \in \B$ _{Row 5, Brown Patch}

$u_{6\brownbox} \in \B$ _{Row 6, Brown Patch}

$v_{1\yellowbox} \in \B$ _{Column 1, Yellow Patch}

$v_{2\yellowbox} \in \B$ _{Column 2, Yellow Patch}

$v_{3\yellowbox} \in \B$ _{Column 3, Yellow Patch}

$v_{4\yellowbox} \in \B$ _{Column 4, Yellow Patch}

$v_{5\yellowbox} \in \B$ _{Column 5, Yellow Patch}

$v_{6\yellowbox} \in \B$ _{Column 6, Yellow Patch}

$v_{1\tealbox} \in \B$ _{Column 1, Tealbox Patch}

$v_{2\tealbox} \in \B$ _{Column 2, Tealbox Patch}

$v_{3\tealbox} \in \B$ _{Column 3, Tealbox Patch}

$v_{4\tealbox} \in \B$ _{Column 4, Tealbox Patch}

$v_{5\tealbox} \in \B$ _{Column 5, Tealbox Patch}

$v_{6\tealbox} \in \B$ _{Column 6, Tealbox Patch}

$v_{1\purplebox} \in \B$ _{Column 1, Purple Patch}

$v_{2\purplebox} \in \B$ _{Column 2, Purple Patch}

$v_{3\purplebox} \in \B$ _{Column 3, Purple Patch}

$v_{4\purplebox} \in \B$ _{Column 4, Purple Patch}

$v_{5\purplebox} \in \B$ _{Column 5, Purple Patch}

$v_{6\purplebox} \in \B$ _{Column 6, Purple Patch}

$v_{1\greenbox} \in \B$ _{Column 1, Green Patch}

$v_{2\greenbox} \in \B$ _{Column 2, Green Patch}

$v_{3\greenbox} \in \B$ _{Column 3, Green Patch}

$v_{4\greenbox} \in \B$ _{Column 4, Green Patch}

$v_{5\greenbox} \in \B$ _{Column 5, Green Patch}

$v_{6\greenbox} \in \B$ _{Column 6, Green Patch}

$v_{1\orangebox} \in \B$ _{Column 1, Orange Patch}

$v_{2\orangebox} \in \B$ _{Column 2, Orange Patch}

$v_{3\orangebox} \in \B$ _{Column 3, Orange Patch}

$v_{4\orangebox} \in \B$ _{Column 4, Orange Patch}

$v_{5\orangebox} \in \B$ _{Column 5, Orange Patch}

$v_{6\orangebox} \in \B$ _{Column 6, Orange Patch}

$v_{1\redbox} \in \B$ _{Column 1, Red Patch}

$v_{2\redbox} \in \B$ _{Column 2, Red Patch}

$v_{3\redbox} \in \B$ _{Column 3, Red Patch}

$v_{4\redbox} \in \B$ _{Column 4, Red Patch}

$v_{5\redbox} \in \B$ _{Column 5, Red Patch}

$v_{6\redbox} \in \B$ _{Column 6, Red Patch}

$v_{1\bluebox} \in \B$ _{Column 1, Blue Patch}

$v_{2\bluebox} \in \B$ _{Column 2, Blue Patch}

$v_{3\bluebox} \in \B$ _{Column 3, Blue Patch}

$v_{4\bluebox} \in \B$ _{Column 4, Blue Patch}

$v_{5\bluebox} \in \B$ _{Column 5, Blue Patch}

$v_{6\bluebox} \in \B$ _{Column 6, Blue Patch}

$v_{1\magentabox} \in \B$ _{Column 1, Magenta Patch}

$v_{2\magentabox} \in \B$ _{Column 2, Magenta Patch}

$v_{3\magentabox} \in \B$ _{Column 3, Magenta Patch}

$v_{4\magentabox} \in \B$ _{Column 4, Magenta Patch}

$v_{5\magentabox} \in \B$ _{Column 5, Magenta Patch}

$v_{6\magentabox} \in \B$ _{Column 6, Magenta Patch}

$v_{1\brickbox} \in \B$ _{Column 1, Brick Patch}

$v_{2\brickbox} \in \B$ _{Column 2, Brick Patch}

$v_{3\brickbox} \in \B$ _{Column 3, Brick Patch}

$v_{4\brickbox} \in \B$ _{Column 4, Brick Patch}

$v_{5\brickbox} \in \B$ _{Column 5, Brick Patch}

$v_{6\brickbox} \in \B$ _{Column 6, Brick Patch}

$v_{1\brownbox} \in \B$ _{Column 1, Brown Patch}

$v_{2\brownbox} \in \B$ _{Column 2, Brown Patch}

$v_{3\brownbox} \in \B$ _{Column 3, Brown Patch}

$v_{4\brownbox} \in \B$ _{Column 4, Brown Patch}

$v_{5\brownbox} \in \B$ _{Column 5, Brown Patch}

$v_{6\brownbox} \in \B$ _{Column 6, Brown Patch}

$x_{11\yellowbox} \in \B$ _{Square (1,1), Yellow Patch}

$x_{12\yellowbox} \in \B$ _{Square (1,2), Yellow Patch}

$x_{13\yellowbox} \in \B$ _{Square (1,3), Yellow Patch}

$x_{14\yellowbox} \in \B$ _{Square (1,4), Yellow Patch}

$x_{15\yellowbox} \in \B$ _{Square (1,5), Yellow Patch}

$x_{16\yellowbox} \in \B$ _{Square (1,6), Yellow Patch}

$x_{21\yellowbox} \in \B$ _{Square (2,1), Yellow Patch}

$x_{22\yellowbox} \in \B$ _{Square (2,2), Yellow Patch}

$x_{23\yellowbox} \in \B$ _{Square (2,3), Yellow Patch}

$x_{24\yellowbox} \in \B$ _{Square (2,4), Yellow Patch}

$x_{25\yellowbox} \in \B$ _{Square (2,5), Yellow Patch}

$x_{26\yellowbox} \in \B$ _{Square (2,6), Yellow Patch}

$x_{31\yellowbox} \in \B$ _{Square (3,1), Yellow Patch}

$x_{32\yellowbox} \in \B$ _{Square (3,2), Yellow Patch}

$x_{33\yellowbox} \in \B$ _{Square (3,3), Yellow Patch}

$x_{34\yellowbox} \in \B$ _{Square (3,4), Yellow Patch}

$x_{35\yellowbox} \in \B$ _{Square (3,5), Yellow Patch}

$x_{36\yellowbox} \in \B$ _{Square (3,6), Yellow Patch}

$x_{41\yellowbox} \in \B$ _{Square (4,1), Yellow Patch}

$x_{42\yellowbox} \in \B$ _{Square (4,2), Yellow Patch}

$x_{43\yellowbox} \in \B$ _{Square (4,3), Yellow Patch}

$x_{44\yellowbox} \in \B$ _{Square (4,4), Yellow Patch}

$x_{45\yellowbox} \in \B$ _{Square (4,5), Yellow Patch}

$x_{46\yellowbox} \in \B$ _{Square (4,6), Yellow Patch}

$x_{51\yellowbox} \in \B$ _{Square (5,1), Yellow Patch}

$x_{52\yellowbox} \in \B$ _{Square (5,2), Yellow Patch}

$x_{53\yellowbox} \in \B$ _{Square (5,3), Yellow Patch}

$x_{54\yellowbox} \in \B$ _{Square (5,4), Yellow Patch}

$x_{55\yellowbox} \in \B$ _{Square (5,5), Yellow Patch}

$x_{56\yellowbox} \in \B$ _{Square (5,6), Yellow Patch}

$x_{61\yellowbox} \in \B$ _{Square (6,1), Yellow Patch}

$x_{62\yellowbox} \in \B$ _{Square (6,2), Yellow Patch}

$x_{63\yellowbox} \in \B$ _{Square (6,3), Yellow Patch}

$x_{64\yellowbox} \in \B$ _{Square (6,4), Yellow Patch}

$x_{65\yellowbox} \in \B$ _{Square (6,5), Yellow Patch}

$x_{66\yellowbox} \in \B$ _{Square (6,6), Yellow Patch}

$x_{11\tealbox} \in \B$ _{Square (1,1), Teal}

$x_{12\tealbox} \in \B$ _{Square (1,2), Teal}

$x_{13\tealbox} \in \B$ _{Square (1,3), Teal}

$x_{14\tealbox} \in \B$ _{Square (1,4), Teal}

$x_{15\tealbox} \in \B$ _{Square (1,5), Teal}

$x_{16\tealbox} \in \B$ _{Square (1,6), Teal}

$x_{21\tealbox} \in \B$ _{Square (2,1), Teal}

$x_{22\tealbox} \in \B$ _{Square (2,2), Teal}

$x_{23\tealbox} \in \B$ _{Square (2,3), Teal}

$x_{24\tealbox} \in \B$ _{Square (2,4), Teal}

$x_{25\tealbox} \in \B$ _{Square (2,5), Teal}

$x_{26\tealbox} \in \B$ _{Square (2,6), Teal}

$x_{31\tealbox} \in \B$ _{Square (3,1), Teal}

$x_{32\tealbox} \in \B$ _{Square (3,2), Teal}

$x_{33\tealbox} \in \B$ _{Square (3,3), Teal}

$x_{34\tealbox} \in \B$ _{Square (3,4), Teal}

$x_{35\tealbox} \in \B$ _{Square (3,5), Teal}

$x_{36\tealbox} \in \B$ _{Square (3,6), Teal}

$x_{41\tealbox} \in \B$ _{Square (4,1), Teal}

$x_{42\tealbox} \in \B$ _{Square (4,2), Teal}

$x_{43\tealbox} \in \B$ _{Square (4,3), Teal}

$x_{44\tealbox} \in \B$ _{Square (4,4), Teal}

$x_{45\tealbox} \in \B$ _{Square (4,5), Teal}

$x_{46\tealbox} \in \B$ _{Square (4,6), Teal}

$x_{51\tealbox} \in \B$ _{Square (5,1), Teal}

$x_{52\tealbox} \in \B$ _{Square (5,2), Teal}

$x_{53\tealbox} \in \B$ _{Square (5,3), Teal}

$x_{54\tealbox} \in \B$ _{Square (5,4), Teal}

$x_{55\tealbox} \in \B$ _{Square (5,5), Teal}

$x_{56\tealbox} \in \B$ _{Square (5,6), Teal}

$x_{61\tealbox} \in \B$ _{Square (6,1), Teal}

$x_{62\tealbox} \in \B$ _{Square (6,2), Teal}

$x_{63\tealbox} \in \B$ _{Square (6,3), Teal}

$x_{64\tealbox} \in \B$ _{Square (6,4), Teal}

$x_{65\tealbox} \in \B$ _{Square (6,5), Teal}

$x_{66\tealbox} \in \B$ _{Square (6,6), Teal}

$x_{11\purplebox} \in \B$ _{Square (1,1), Purple Patch}

$x_{12\purplebox} \in \B$ _{Square (1,2), Purple Patch}

$x_{13\purplebox} \in \B$ _{Square (1,3), Purple Patch}

$x_{14\purplebox} \in \B$ _{Square (1,4), Purple Patch}

$x_{15\purplebox} \in \B$ _{Square (1,5), Purple Patch}

$x_{16\purplebox} \in \B$ _{Square (1,6), Purple Patch}

$x_{21\purplebox} \in \B$ _{Square (2,1), Purple Patch}

$x_{22\purplebox} \in \B$ _{Square (2,2), Purple Patch}

$x_{23\purplebox} \in \B$ _{Square (2,3), Purple Patch}

$x_{24\purplebox} \in \B$ _{Square (2,4), Purple Patch}

$x_{25\purplebox} \in \B$ _{Square (2,5), Purple Patch}

$x_{26\purplebox} \in \B$ _{Square (2,6), Purple Patch}

$x_{31\purplebox} \in \B$ _{Square (3,1), Purple Patch}

$x_{32\purplebox} \in \B$ _{Square (3,2), Purple Patch}

$x_{33\purplebox} \in \B$ _{Square (3,3), Purple Patch}

$x_{34\purplebox} \in \B$ _{Square (3,4), Purple Patch}

$x_{35\purplebox} \in \B$ _{Square (3,5), Purple Patch}

$x_{36\purplebox} \in \B$ _{Square (3,6), Purple Patch}

$x_{41\purplebox} \in \B$ _{Square (4,1), Purple Patch}

$x_{42\purplebox} \in \B$ _{Square (4,2), Purple Patch}

$x_{43\purplebox} \in \B$ _{Square (4,3), Purple Patch}

$x_{44\purplebox} \in \B$ _{Square (4,4), Purple Patch}

$x_{45\purplebox} \in \B$ _{Square (4,5), Purple Patch}

$x_{46\purplebox} \in \B$ _{Square (4,6), Purple Patch}

$x_{51\purplebox} \in \B$ _{Square (5,1), Purple Patch}

$x_{52\purplebox} \in \B$ _{Square (5,2), Purple Patch}

$x_{53\purplebox} \in \B$ _{Square (5,3), Purple Patch}

$x_{54\purplebox} \in \B$ _{Square (5,4), Purple Patch}

$x_{55\purplebox} \in \B$ _{Square (5,5), Purple Patch}

$x_{56\purplebox} \in \B$ _{Square (5,6), Purple Patch}

$x_{61\purplebox} \in \B$ _{Square (6,1), Purple Patch}

$x_{62\purplebox} \in \B$ _{Square (6,2), Purple Patch}

$x_{63\purplebox} \in \B$ _{Square (6,3), Purple Patch}

$x_{64\purplebox} \in \B$ _{Square (6,4), Purple Patch}

$x_{65\purplebox} \in \B$ _{Square (6,5), Purple Patch}

$x_{66\purplebox} \in \B$ _{Square (6,6), Purple Patch}

$x_{11\greenbox} \in \B$ _{Square (1,1), Green Patch}

$x_{12\greenbox} \in \B$ _{Square (1,2), Green Patch}

$x_{13\greenbox} \in \B$ _{Square (1,3), Green Patch}

$x_{14\greenbox} \in \B$ _{Square (1,4), Green Patch}

$x_{15\greenbox} \in \B$ _{Square (1,5), Green Patch}

$x_{16\greenbox} \in \B$ _{Square (1,6), Green Patch}

$x_{21\greenbox} \in \B$ _{Square (2,1), Green Patch}

$x_{22\greenbox} \in \B$ _{Square (2,2), Green Patch}

$x_{23\greenbox} \in \B$ _{Square (2,3), Green Patch}

$x_{24\greenbox} \in \B$ _{Square (2,4), Green Patch}

$x_{25\greenbox} \in \B$ _{Square (2,5), Green Patch}

$x_{26\greenbox} \in \B$ _{Square (2,6), Green Patch}

$x_{31\greenbox} \in \B$ _{Square (3,1), Green Patch}

$x_{32\greenbox} \in \B$ _{Square (3,2), Green Patch}

$x_{33\greenbox} \in \B$ _{Square (3,3), Green Patch}

$x_{34\greenbox} \in \B$ _{Square (3,4), Green Patch}

$x_{35\greenbox} \in \B$ _{Square (3,5), Green Patch}

$x_{36\greenbox} \in \B$ _{Square (3,6), Green Patch}

$x_{41\greenbox} \in \B$ _{Square (4,1), Green Patch}

$x_{42\greenbox} \in \B$ _{Square (4,2), Green Patch}

$x_{43\greenbox} \in \B$ _{Square (4,3), Green Patch}

$x_{44\greenbox} \in \B$ _{Square (4,4), Green Patch}

$x_{45\greenbox} \in \B$ _{Square (4,5), Green Patch}

$x_{46\greenbox} \in \B$ _{Square (4,6), Green Patch}

$x_{51\greenbox} \in \B$ _{Square (5,1), Green Patch}

$x_{52\greenbox} \in \B$ _{Square (5,2), Green Patch}

$x_{53\greenbox} \in \B$ _{Square (5,3), Green Patch}

$x_{54\greenbox} \in \B$ _{Square (5,4), Green Patch}

$x_{55\greenbox} \in \B$ _{Square (5,5), Green Patch}

$x_{56\greenbox} \in \B$ _{Square (5,6), Green Patch}

$x_{61\greenbox} \in \B$ _{Square (6,1), Green Patch}

$x_{62\greenbox} \in \B$ _{Square (6,2), Green Patch}

$x_{63\greenbox} \in \B$ _{Square (6,3), Green Patch}

$x_{64\greenbox} \in \B$ _{Square (6,4), Green Patch}

$x_{65\greenbox} \in \B$ _{Square (6,5), Green Patch}

$x_{66\greenbox} \in \B$ _{Square (6,6), Green Patch}

$x_{11\orangebox} \in \B$ _{Square (1,1), Orange Patch}

$x_{12\orangebox} \in \B$ _{Square (1,2), Orange Patch}

$x_{13\orangebox} \in \B$ _{Square (1,3), Orange Patch}

$x_{14\orangebox} \in \B$ _{Square (1,4), Orange Patch}

$x_{15\orangebox} \in \B$ _{Square (1,5), Orange Patch}

$x_{16\orangebox} \in \B$ _{Square (1,6), Orange Patch}

$x_{21\orangebox} \in \B$ _{Square (2,1), Orange Patch}

$x_{22\orangebox} \in \B$ _{Square (2,2), Orange Patch}

$x_{23\orangebox} \in \B$ _{Square (2,3), Orange Patch}

$x_{24\orangebox} \in \B$ _{Square (2,4), Orange Patch}

$x_{25\orangebox} \in \B$ _{Square (2,5), Orange Patch}

$x_{26\orangebox} \in \B$ _{Square (2,6), Orange Patch}

$x_{31\orangebox} \in \B$ _{Square (3,1), Orange Patch}

$x_{32\orangebox} \in \B$ _{Square (3,2), Orange Patch}

$x_{33\orangebox} \in \B$ _{Square (3,3), Orange Patch}

$x_{34\orangebox} \in \B$ _{Square (3,4), Orange Patch}

$x_{35\orangebox} \in \B$ _{Square (3,5), Orange Patch}

$x_{36\orangebox} \in \B$ _{Square (3,6), Orange Patch}

$x_{41\orangebox} \in \B$ _{Square (4,1), Orange Patch}

$x_{42\orangebox} \in \B$ _{Square (4,2), Orange Patch}

$x_{43\orangebox} \in \B$ _{Square (4,3), Orange Patch}

$x_{44\orangebox} \in \B$ _{Square (4,4), Orange Patch}

$x_{45\orangebox} \in \B$ _{Square (4,5), Orange Patch}

$x_{46\orangebox} \in \B$ _{Square (4,6), Orange Patch}

$x_{51\orangebox} \in \B$ _{Square (5,1), Orange Patch}

$x_{52\orangebox} \in \B$ _{Square (5,2), Orange Patch}

$x_{53\orangebox} \in \B$ _{Square (5,3), Orange Patch}

$x_{54\orangebox} \in \B$ _{Square (5,4), Orange Patch}

$x_{55\orangebox} \in \B$ _{Square (5,5), Orange Patch}

$x_{56\orangebox} \in \B$ _{Square (5,6), Orange Patch}

$x_{61\orangebox} \in \B$ _{Square (6,1), Orange Patch}

$x_{62\orangebox} \in \B$ _{Square (6,2), Orange Patch}

$x_{63\orangebox} \in \B$ _{Square (6,3), Orange Patch}

$x_{64\orangebox} \in \B$ _{Square (6,4), Orange Patch}

$x_{65\orangebox} \in \B$ _{Square (6,5), Orange Patch}

$x_{66\orangebox} \in \B$ _{Square (6,6), Orange Patch}

$x_{11\redbox} \in \B$ _{Square (1,1), Red Patch}

$x_{12\redbox} \in \B$ _{Square (1,2), Red Patch}

$x_{13\redbox} \in \B$ _{Square (1,3), Red Patch}

$x_{14\redbox} \in \B$ _{Square (1,4), Red Patch}

$x_{15\redbox} \in \B$ _{Square (1,5), Red Patch}

$x_{16\redbox} \in \B$ _{Square (1,6), Red Patch}

$x_{21\redbox} \in \B$ _{Square (2,1), Red Patch}

$x_{22\redbox} \in \B$ _{Square (2,2), Red Patch}

$x_{23\redbox} \in \B$ _{Square (2,3), Red Patch}

$x_{24\redbox} \in \B$ _{Square (2,4), Red Patch}

$x_{25\redbox} \in \B$ _{Square (2,5), Red Patch}

$x_{26\redbox} \in \B$ _{Square (2,6), Red Patch}

$x_{31\redbox} \in \B$ _{Square (3,1), Red Patch}

$x_{32\redbox} \in \B$ _{Square (3,2), Red Patch}

$x_{33\redbox} \in \B$ _{Square (3,3), Red Patch}

$x_{34\redbox} \in \B$ _{Square (3,4), Red Patch}

$x_{35\redbox} \in \B$ _{Square (3,5), Red Patch}

$x_{36\redbox} \in \B$ _{Square (3,6), Red Patch}

$x_{41\redbox} \in \B$ _{Square (4,1), Red Patch}

$x_{42\redbox} \in \B$ _{Square (4,2), Red Patch}

$x_{43\redbox} \in \B$ _{Square (4,3), Red Patch}

$x_{44\redbox} \in \B$ _{Square (4,4), Red Patch}

$x_{45\redbox} \in \B$ _{Square (4,5), Red Patch}

$x_{46\redbox} \in \B$ _{Square (4,6), Red Patch}

$x_{51\redbox} \in \B$ _{Square (5,1), Red Patch}

$x_{52\redbox} \in \B$ _{Square (5,2), Red Patch}

$x_{53\redbox} \in \B$ _{Square (5,3), Red Patch}

$x_{54\redbox} \in \B$ _{Square (5,4), Red Patch}

$x_{55\redbox} \in \B$ _{Square (5,5), Red Patch}

$x_{56\redbox} \in \B$ _{Square (5,6), Red Patch}

$x_{61\redbox} \in \B$ _{Square (6,1), Red Patch}

$x_{62\redbox} \in \B$ _{Square (6,2), Red Patch}

$x_{63\redbox} \in \B$ _{Square (6,3), Red Patch}

$x_{64\redbox} \in \B$ _{Square (6,4), Red Patch}

$x_{65\redbox} \in \B$ _{Square (6,5), Red Patch}

$x_{66\redbox} \in \B$ _{Square (6,6), Red Patch}

$x_{11\bluebox} \in \B$ _{Square (1,1), Blue Patch}

$x_{12\bluebox} \in \B$ _{Square (1,2), Blue Patch}

$x_{13\bluebox} \in \B$ _{Square (1,3), Blue Patch}

$x_{14\bluebox} \in \B$ _{Square (1,4), Blue Patch}

$x_{15\bluebox} \in \B$ _{Square (1,5), Blue Patch}

$x_{16\bluebox} \in \B$ _{Square (1,6), Blue Patch}

$x_{21\bluebox} \in \B$ _{Square (2,1), Blue Patch}

$x_{22\bluebox} \in \B$ _{Square (2,2), Blue Patch}

$x_{23\bluebox} \in \B$ _{Square (2,3), Blue Patch}

$x_{24\bluebox} \in \B$ _{Square (2,4), Blue Patch}

$x_{25\bluebox} \in \B$ _{Square (2,5), Blue Patch}

$x_{26\bluebox} \in \B$ _{Square (2,6), Blue Patch}

$x_{31\bluebox} \in \B$ _{Square (3,1), Blue Patch}

$x_{32\bluebox} \in \B$ _{Square (3,2), Blue Patch}

$x_{33\bluebox} \in \B$ _{Square (3,3), Blue Patch}

$x_{34\bluebox} \in \B$ _{Square (3,4), Blue Patch}

$x_{35\bluebox} \in \B$ _{Square (3,5), Blue Patch}

$x_{36\bluebox} \in \B$ _{Square (3,6), Blue Patch}

$x_{41\bluebox} \in \B$ _{Square (4,1), Blue Patch}

$x_{42\bluebox} \in \B$ _{Square (4,2), Blue Patch}

$x_{43\bluebox} \in \B$ _{Square (4,3), Blue Patch}

$x_{44\bluebox} \in \B$ _{Square (4,4), Blue Patch}

$x_{45\bluebox} \in \B$ _{Square (4,5), Blue Patch}

$x_{46\bluebox} \in \B$ _{Square (4,6), Blue Patch}

$x_{51\bluebox} \in \B$ _{Square (5,1), Blue Patch}

$x_{52\bluebox} \in \B$ _{Square (5,2), Blue Patch}

$x_{53\bluebox} \in \B$ _{Square (5,3), Blue Patch}

$x_{54\bluebox} \in \B$ _{Square (5,4), Blue Patch}

$x_{55\bluebox} \in \B$ _{Square (5,5), Blue Patch}

$x_{56\bluebox} \in \B$ _{Square (5,6), Blue Patch}

$x_{61\bluebox} \in \B$ _{Square (6,1), Blue Patch}

$x_{62\bluebox} \in \B$ _{Square (6,2), Blue Patch}

$x_{63\bluebox} \in \B$ _{Square (6,3), Blue Patch}

$x_{64\bluebox} \in \B$ _{Square (6,4), Blue Patch}

$x_{65\bluebox} \in \B$ _{Square (6,5), Blue Patch}

$x_{66\bluebox} \in \B$ _{Square (6,6), Blue Patch}

$x_{11\magentabox} \in \B$ _{Square (1,1), Magenta Patch}

$x_{12\magentabox} \in \B$ _{Square (1,2), Magenta Patch}

$x_{13\magentabox} \in \B$ _{Square (1,3), Magenta Patch}

$x_{14\magentabox} \in \B$ _{Square (1,4), Magenta Patch}

$x_{15\magentabox} \in \B$ _{Square (1,5), Magenta Patch}

$x_{16\magentabox} \in \B$ _{Square (1,6), Magenta Patch}

$x_{21\magentabox} \in \B$ _{Square (2,1), Magenta Patch}

$x_{22\magentabox} \in \B$ _{Square (2,2), Magenta Patch}

$x_{23\magentabox} \in \B$ _{Square (2,3), Magenta Patch}

$x_{24\magentabox} \in \B$ _{Square (2,4), Magenta Patch}

$x_{25\magentabox} \in \B$ _{Square (2,5), Magenta Patch}

$x_{26\magentabox} \in \B$ _{Square (2,6), Magenta Patch}

$x_{31\magentabox} \in \B$ _{Square (3,1), Magenta Patch}

$x_{32\magentabox} \in \B$ _{Square (3,2), Magenta Patch}

$x_{33\magentabox} \in \B$ _{Square (3,3), Magenta Patch}

$x_{34\magentabox} \in \B$ _{Square (3,4), Magenta Patch}

$x_{35\magentabox} \in \B$ _{Square (3,5), Magenta Patch}

$x_{36\magentabox} \in \B$ _{Square (3,6), Magenta Patch}

$x_{41\magentabox} \in \B$ _{Square (4,1), Magenta Patch}

$x_{42\magentabox} \in \B$ _{Square (4,2), Magenta Patch}

$x_{43\magentabox} \in \B$ _{Square (4,3), Magenta Patch}

$x_{44\magentabox} \in \B$ _{Square (4,4), Magenta Patch}

$x_{45\magentabox} \in \B$ _{Square (4,5), Magenta Patch}

$x_{46\magentabox} \in \B$ _{Square (4,6), Magenta Patch}

$x_{51\magentabox} \in \B$ _{Square (5,1), Magenta Patch}

$x_{52\magentabox} \in \B$ _{Square (5,2), Magenta Patch}

$x_{53\magentabox} \in \B$ _{Square (5,3), Magenta Patch}

$x_{54\magentabox} \in \B$ _{Square (5,4), Magenta Patch}

$x_{55\magentabox} \in \B$ _{Square (5,5), Magenta Patch}

$x_{56\magentabox} \in \B$ _{Square (5,6), Magenta Patch}

$x_{61\magentabox} \in \B$ _{Square (6,1), Magenta Patch}

$x_{62\magentabox} \in \B$ _{Square (6,2), Magenta Patch}

$x_{63\magentabox} \in \B$ _{Square (6,3), Magenta Patch}

$x_{64\magentabox} \in \B$ _{Square (6,4), Magenta Patch}

$x_{65\magentabox} \in \B$ _{Square (6,5), Magenta Patch}

$x_{66\magentabox} \in \B$ _{Square (6,6), Magenta Patch}

$x_{11\brickbox} \in \B$ _{Square (1,1), Brick Patch}

$x_{12\brickbox} \in \B$ _{Square (1,2), Brick Patch}

$x_{13\brickbox} \in \B$ _{Square (1,3), Brick Patch}

$x_{14\brickbox} \in \B$ _{Square (1,4), Brick Patch}

$x_{15\brickbox} \in \B$ _{Square (1,5), Brick Patch}

$x_{16\brickbox} \in \B$ _{Square (1,6), Brick Patch}

$x_{21\brickbox} \in \B$ _{Square (2,1), Brick Patch}

$x_{22\brickbox} \in \B$ _{Square (2,2), Brick Patch}

$x_{23\brickbox} \in \B$ _{Square (2,3), Brick Patch}

$x_{24\brickbox} \in \B$ _{Square (2,4), Brick Patch}

$x_{25\brickbox} \in \B$ _{Square (2,5), Brick Patch}

$x_{26\brickbox} \in \B$ _{Square (2,6), Brick Patch}

$x_{31\brickbox} \in \B$ _{Square (3,1), Brick Patch}

$x_{32\brickbox} \in \B$ _{Square (3,2), Brick Patch}

$x_{33\brickbox} \in \B$ _{Square (3,3), Brick Patch}

$x_{34\brickbox} \in \B$ _{Square (3,4), Brick Patch}

$x_{35\brickbox} \in \B$ _{Square (3,5), Brick Patch}

$x_{36\brickbox} \in \B$ _{Square (3,6), Brick Patch}

$x_{41\brickbox} \in \B$ _{Square (4,1), Brick Patch}

$x_{42\brickbox} \in \B$ _{Square (4,2), Brick Patch}

$x_{43\brickbox} \in \B$ _{Square (4,3), Brick Patch}

$x_{44\brickbox} \in \B$ _{Square (4,4), Brick Patch}

$x_{45\brickbox} \in \B$ _{Square (4,5), Brick Patch}

$x_{46\brickbox} \in \B$ _{Square (4,6), Brick Patch}

$x_{51\brickbox} \in \B$ _{Square (5,1), Brick Patch}

$x_{52\brickbox} \in \B$ _{Square (5,2), Brick Patch}

$x_{53\brickbox} \in \B$ _{Square (5,3), Brick Patch}

$x_{54\brickbox} \in \B$ _{Square (5,4), Brick Patch}

$x_{55\brickbox} \in \B$ _{Square (5,5), Brick Patch}

$x_{56\brickbox} \in \B$ _{Square (5,6), Brick Patch}

$x_{61\brickbox} \in \B$ _{Square (6,1), Brick Patch}

$x_{62\brickbox} \in \B$ _{Square (6,2), Brick Patch}

$x_{63\brickbox} \in \B$ _{Square (6,3), Brick Patch}

$x_{64\brickbox} \in \B$ _{Square (6,4), Brick Patch}

$x_{65\brickbox} \in \B$ _{Square (6,5), Brick Patch}

$x_{66\brickbox} \in \B$ _{Square (6,6), Brick Patch}

$x_{11\brownbox} \in \B$ _{Square (1,1), Brown Patch}

$x_{12\brownbox} \in \B$ _{Square (1,2), Brown Patch}

$x_{13\brownbox} \in \B$ _{Square (1,3), Brown Patch}

$x_{14\brownbox} \in \B$ _{Square (1,4), Brown Patch}

$x_{15\brownbox} \in \B$ _{Square (1,5), Brown Patch}

$x_{16\brownbox} \in \B$ _{Square (1,6), Brown Patch}

$x_{21\brownbox} \in \B$ _{Square (2,1), Brown Patch}

$x_{22\brownbox} \in \B$ _{Square (2,2), Brown Patch}

$x_{23\brownbox} \in \B$ _{Square (2,3), Brown Patch}

$x_{24\brownbox} \in \B$ _{Square (2,4), Brown Patch}

$x_{25\brownbox} \in \B$ _{Square (2,5), Brown Patch}

$x_{26\brownbox} \in \B$ _{Square (2,6), Brown Patch}

$x_{31\brownbox} \in \B$ _{Square (3,1), Brown Patch}

$x_{32\brownbox} \in \B$ _{Square (3,2), Brown Patch}

$x_{33\brownbox} \in \B$ _{Square (3,3), Brown Patch}

$x_{34\brownbox} \in \B$ _{Square (3,4), Brown Patch}

$x_{35\brownbox} \in \B$ _{Square (3,5), Brown Patch}

$x_{36\brownbox} \in \B$ _{Square (3,6), Brown Patch}

$x_{41\brownbox} \in \B$ _{Square (4,1), Brown Patch}

$x_{42\brownbox} \in \B$ _{Square (4,2), Brown Patch}

$x_{43\brownbox} \in \B$ _{Square (4,3), Brown Patch}

$x_{44\brownbox} \in \B$ _{Square (4,4), Brown Patch}

$x_{45\brownbox} \in \B$ _{Square (4,5), Brown Patch}

$x_{46\brownbox} \in \B$ _{Square (4,6), Brown Patch}

$x_{51\brownbox} \in \B$ _{Square (5,1), Brown Patch}

$x_{52\brownbox} \in \B$ _{Square (5,2), Brown Patch}

$x_{53\brownbox} \in \B$ _{Square (5,3), Brown Patch}

$x_{54\brownbox} \in \B$ _{Square (5,4), Brown Patch}

$x_{55\brownbox} \in \B$ _{Square (5,5), Brown Patch}

$x_{56\brownbox} \in \B$ _{Square (5,6), Brown Patch}

$x_{61\brownbox} \in \B$ _{Square (6,1), Brown Patch}

$x_{62\brownbox} \in \B$ _{Square (6,2), Brown Patch}

$x_{63\brownbox} \in \B$ _{Square (6,3), Brown Patch}

$x_{64\brownbox} \in \B$ _{Square (6,4), Brown Patch}

$x_{65\brownbox} \in \B$ _{Square (6,5), Brown Patch}

$x_{66\brownbox} \in \B$ _{Square (6,6), Brown Patch}