Magnet Puzzle

The puzzle game Magnets involves placing a set of domino-shaped magnets (or electrets or other polarized objects) in a subset of slots on a board so as to satisfy a set of constraints. For example, the puzzle on the left has the solution shown on the right: 👁 Image

Each slot contains either a blank entry (indicated by ‘x’s), or a “magnet” with a positive and negative end. The numbers along the left and top sides show the numbers of ‘+’ squares in particular rows or columns. Those along the right and bottom show the number of ‘-’ signs in particular rows or columns. Rows and columns without a number at one or both ends are unconstrained as to the number of ‘+’ or ‘-’ signs, depending on which number is not present. In addition to fulfilling these numerical constraints, a puzzle solution must also satisfy the constraint that no two orthogonally touching squares may have the same sign (diagonally joined squares are not constrained). You are given top[], bottom[], left[], right[] arrays indicates the count of + or - along the top(+), bottom(-), left(+) and right(-) edges respectively. Values of -1 indicate any number of + and - signs. Also given matrix rules[][] contain any one T, B, L or R characters. For a vertical slot in the board, T indicates its top end and B indicates the bottom end. For a horizontal slot in the board, L indicates left end and R indicates the right end. 

Examples:

Input : M = 5, N = 6
 top[] = { 1, -1, -1, 2, 1, -1 }
 bottom[] = { 2, -1, -1, 2, -1, 3 }
 left[] = { 2, 3, -1, -1, -1 }
 right[] = { -1, -1, -1, 1, -1 }
 rules[][] = { { L, R, L, R, T, T },
 { L, R, L, R, B, B },
 { T, T, T, T, L, R },
 { B, B, B, B, T, T },
 { L, R, L, R, B, B }};
Output : + - + - X - 
 - + - + X + 
 X X + - + - 
 X X - + X + 
 - + X X X - 

Input : M = 4, N = 3
 top[] = { 2, -1, -1 }
 bottom[] = { -1, -1, 2 }
 left[] = { -1, -1, 2, -1 }
 right[] = { 0, -1, -1, -1 }
 rules[][] = { { T, T, T },
 { B, B, B },
 { T, L, R },
 { B, L, R } };
Output : + X +
 – X –
 + – +
 – + –

We can solve this problem using Backtracking. 

Output: 
[['+', '-', '+', '-', 'x', '-', ]['-', '+', '-', '+', 'x', '+', ]['x', 'x', '+', '-', '+', '-', ]['x', 'x', '-', '+', 'x', '+', ]['-', '+', 'x', 'x', 'x', '-', ]]

Source :https://people.eecs.berkeley.edu/~hilfingr/programming-contest/f2012-contest.pdf