[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (1, 1) is white, cell (2, 1) is white, cell (3, 0) is white, cell (0, 3) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}

dataset: MATH


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 1) is black, cell (3, 2) is black, cell (2, 3) is white, cell (0, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is black, cell (3, 0) is black, cell (2, 3) is black, cell (2, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 4) is white, cell (0, 1) is black, cell (3, 2) is white, cell (1, 3) is white, cell (1, 0) is white, cell (2, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is white, cell (1, 0) is white, cell (2, 1) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 0) is black, cell (3, 1) is white, cell (3, 2) is white, cell (1, 0) is black, cell (0, 0) is black, cell (3, 3) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is white, cell (1, 2) is white, cell (4, 0) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is white, cell (2, 1) is white, cell (1, 2) is white, cell (0, 0) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (4, 4) is black, cell (1, 0) is black, cell (2, 1) is black, cell (0, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 2) is white, cell (0, 3) is black, cell (2, 3) is black, cell (3, 1) is white, cell (1, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is white, cell (0, 3) is white, cell (4, 1) is white, cell (2, 4) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is white, cell (2, 1) is black, cell (3, 2) is white, cell (1, 4) is black, cell (4, 0) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is black, cell (2, 2) is black, cell (3, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 4) is black, cell (3, 2) is white, cell (1, 2) is white, cell (2, 3) is black, cell (0, 0) is black, cell (4, 2) is white, cell (2, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (1, 4) is black, cell (2, 2) is black, cell (4, 0) is black, cell (1, 3) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (2, 0) is black, cell (0, 1) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is white, cell (2, 3) is black, cell (3, 4) is white, cell (0, 2) is black, cell (4, 1) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (1, 4) is black, cell (1, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 0) is black, cell (3, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 2) is black, cell (2, 0) is black, cell (1, 2) is black, cell (3, 1) is black, cell (0, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is black, cell (0, 1) is black, cell (2, 3) is black, cell (2, 3) is black, cell (2, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 0) is black, cell (0, 1) is black, cell (1, 2) is white, cell (3, 3) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 2) is black, cell (3, 2) is black, cell (0, 3) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (2, 4) is black, cell (0, 1) is white, cell (4, 4) is black, cell (0, 2) is white, cell (1, 0) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 4) is white, cell (0, 0) is white, cell (0, 1) is white, cell (1, 2) is white, cell (0, 3) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 3) is black, cell (4, 4) is white, cell (3, 0) is black, cell (4, 1) is white, cell (0, 3) is black, cell (1, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is black, cell (2, 2) is white, cell (0, 1) is black, cell (3, 3) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 4) is black, cell (0, 4) is black, cell (3, 0) is black, cell (3, 1) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is white, cell (3, 3) is white, cell (2, 1) is white, cell (0, 4) is white, cell (4, 1) is white, cell (2, 1) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 2) is white, cell (4, 0) is white, cell (2, 0) is white, cell (0, 3) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is black, cell (4, 2) is black, cell (0, 4) is black, cell (1, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 3) is black, cell (1, 3) is black, cell (3, 2) is white, cell (2, 0) is white, cell (0, 1) is black, cell (2, 4) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 4) is white, cell (1, 0) is white, cell (2, 4) is white, cell (4, 1) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is black, cell (1, 3) is black, cell (1, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 1) is black, cell (0, 3) is white, cell (1, 0) is white, cell (0, 2) is white, cell (3, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is white, cell (4, 1) is black, cell (4, 2) is black, cell (1, 3) is white, cell (3, 4) is black, cell (2, 3) is white, cell (4, 0) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 1) is white, cell (4, 0) is white, cell (4, 3) is white, cell (3, 2) is white, cell (3, 2) is white, cell (1, 1) is white, cell (2, 2) is white, cell (3, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (3, 3) is black, cell (2, 1) is black, cell (3, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (2, 2) is black, cell (2, 2) is black, cell (3, 1) is black, cell (1, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is white, cell (2, 2) is white, cell (1, 3) is white, cell (2, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (1, 1) is white, cell (3, 0) is white, cell (2, 3) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is white, cell (2, 2) is black, cell (0, 3) is white, cell (0, 1) is white, cell (3, 2) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 4) is black, cell (0, 3) is white, cell (3, 1) is black, cell (4, 2) is black, cell (2, 2) is black, cell (0, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is white, cell (2, 4) is white, cell (4, 3) is white, cell (3, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (2, 2) is white, cell (0, 0) is white, cell (1, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 0) is white, cell (3, 2) is white, cell (0, 1) is black, cell (1, 3) is black, cell (2, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 3) is white, cell (3, 2) is white, cell (3, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 1) is black, cell (3, 3) is white, cell (3, 2) is white, cell (0, 0) is white, cell (2, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (0, 2) is black, cell (2, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is black, cell (2, 0) is white, cell (1, 2) is black, cell (4, 4) is white, cell (0, 1) is black, cell (3, 4) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is white, cell (1, 0) is white, cell (0, 3) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 4) is white, cell (2, 0) is white, cell (3, 0) is white, cell (0, 0) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 1) is black, cell (0, 2) is black, cell (3, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is white, cell (1, 0) is white, cell (3, 1) is black, cell (0, 1) is black, cell (2, 2) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 4) is black, cell (2, 0) is white, cell (3, 1) is white, cell (0, 3) is white, cell (4, 2) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 1) is white, cell (1, 3) is black, cell (2, 3) is black, cell (0, 0) is white, cell (3, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is black, cell (2, 0) is white, cell (0, 2) is black, cell (0, 1) is black, cell (3, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (2, 1) is white, cell (3, 0) is black, cell (1, 3) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 1) is black, cell (3, 1) is black, cell (2, 2) is black, cell (0, 1) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is black, cell (1, 2) is black, cell (2, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 1) is black, cell (3, 2) is white, cell (1, 3) is white, cell (2, 2) is white, cell (0, 0) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 2) is white, cell (4, 4) is white, cell (2, 3) is white, cell (3, 4) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 1) is black, cell (2, 2) is black, cell (2, 4) is black, cell (3, 3) is black, cell (3, 3) is black, cell (0, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 0) is white, cell (0, 3) is white, cell (3, 2) is black, cell (1, 3) is white, cell (3, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 2) is white, cell (0, 1) is white, cell (1, 2) is white, cell (1, 0) is white, cell (1, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 4) is black, cell (3, 3) is black, cell (1, 0) is black, cell (0, 1) is black, cell (2, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is black, cell (4, 3) is black, cell (1, 4) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 1) is black, cell (1, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 0) is black, cell (0, 4) is white, cell (1, 0) is black, cell (3, 2) is white, cell (3, 1) is white, cell (0, 3) is white, cell (2, 0) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is white, cell (1, 3) is black, cell (0, 3) is black, cell (3, 0) is black, cell (2, 1) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is black, cell (2, 3) is black, cell (3, 1) is black, cell (1, 2) is black, cell (2, 3) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is white, cell (0, 1) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 1) is black, cell (4, 3) is white, cell (1, 3) is white, cell (3, 4) is white, cell (2, 0) is white, cell (0, 2) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is black, cell (1, 1) is black, cell (2, 2) is black, cell (0, 0) is white, cell (3, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is black, cell (2, 4) is white, cell (3, 0) is white, cell (0, 2) is white, cell (4, 2) is white, cell (1, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 4) is black, cell (2, 2) is white, cell (1, 0) is black, cell (3, 3) is white, cell (4, 2) is white, cell (0, 2) is white, cell (1, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 1) is black, cell (1, 3) is white, cell (3, 0) is black, cell (2, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is white, cell (2, 3) is white, cell (3, 0) is black, cell (1, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 3) is white, cell (4, 4) is white, cell (3, 4) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 2) is black, cell (4, 0) is black, cell (1, 3) is black, cell (3, 1) is black, cell (3, 1) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 2) is white, cell (1, 3) is white, cell (0, 0) is white, cell (0, 0) is white, cell (2, 0) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 2) is black, cell (3, 3) is black, cell (0, 2) is black, cell (0, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 0) is black, cell (0, 0) is black, cell (3, 0) is black, cell (3, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 4) is black, cell (2, 3) is black, cell (1, 2) is black, cell (1, 2) is black, cell (0, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 4) is white, cell (3, 4) is white, cell (1, 3) is white, cell (0, 1) is white, cell (4, 2) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 4) is black, cell (0, 1) is black, cell (3, 2) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 4) is black, cell (4, 0) is black, cell (0, 2) is black, cell (3, 4) is black, cell (0, 1) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 1) is black, cell (2, 0) is black, cell (0, 2) is white, cell (3, 3) is black, cell (1, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is black, cell (2, 0) is black, cell (3, 3) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 1) is white, cell (0, 4) is black, cell (1, 3) is black, cell (2, 0) is white, cell (4, 1) is white, cell (0, 2) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 1) is black, cell (4, 0) is black, cell (3, 2) is black, cell (1, 3) is black, cell (1, 3) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (1, 3) is white, cell (0, 2) is white, cell (3, 2) is white, cell (2, 0) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is white, cell (0, 3) is white, cell (2, 4) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 0) is white, cell (0, 4) is white, cell (4, 1) is black, cell (3, 3) is black, cell (2, 3) is black, cell (1, 3) is black, cell (0, 2) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (3, 3) is black, cell (2, 2) is white, cell (1, 2) is white, cell (3, 1) is black, cell (0, 0) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is white, cell (1, 1) is white, cell (0, 1) is white, cell (3, 0) is black, cell (3, 3) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 2) is white, cell (3, 0) is black, cell (1, 3) is black, cell (0, 1) is white. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 3) is white, cell (2, 1) is white, cell (1, 0) is white, cell (4, 3) is white. What is the maximum number of black chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 4) is black, cell (3, 1) is black, cell (4, 3) is black, cell (2, 2) is black, cell (2, 0) is black. What is the maximum number of white chips that can be placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}


[User]
Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (2, 0) is white, cell (3, 1) is black, cell (2, 3) is white, cell (1, 1) is black, cell (2, 2) is white, cell (0, 1) is black. How many chips are placed on the grid?

Present the answer in LaTex format: \boxed{Your answer}