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:

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.

`# Write Python3 code here ` `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"` `]]; ` ` ` ` ` ` ` `def` `canPutPatternHorizontally(rules,i,j,pat): ` ` ` ` ` `if` `j` `-` `1` `>` `=` `0` `and` `rules[i][j` `-` `1` `] ` `=` `=` `pat[` `0` `]: ` ` ` `return` `False` ` ` `elif` `i` `-` `1` `>` `=` `0` `and` `rules[i` `-` `1` `][j] ` `=` `=` `pat[` `0` `]: ` ` ` `return` `False` ` ` `elif` `i` `-` `1` `>` `=` `0` `and` `rules[i` `-` `1` `][j` `+` `1` `] ` `=` `=` `pat[` `1` `]: ` ` ` `return` `False` ` ` `elif` `j` `+` `2` `< ` `len` `(rules[` `0` `]) ` `and` `rules[i][j` `+` `2` `] ` `=` `=` `pat[` `1` `]: ` ` ` `return` `False` ` ` ` ` `return` `True` ` ` ` ` `def` `canPutPatternVertically(rules,i,j,pat): ` ` ` ` ` `if` `j` `-` `1` `>` `=` `0` `and` `rules[i][j` `-` `1` `] ` `=` `=` `pat[` `0` `]: ` ` ` `return` `False` ` ` `elif` `i` `-` `1` `>` `=` `0` `and` `rules[i` `-` `1` `][j] ` `=` `=` `pat[` `0` `]: ` ` ` `return` `False` ` ` `elif` `j` `+` `1` `< ` `len` `(rules[` `0` `]) ` `and` `rules[i][j` `+` `1` `] ` `=` `=` `pat[` `0` `]: ` ` ` `return` `False` ` ` ` ` `return` `True` ` ` `def` `doTheStuff(rules,i,j): ` ` ` ` ` `if` `rules[i][j] ` `=` `=` `"L"` `or` `rules[i][j] ` `=` `=` `"R"` `: ` ` ` ` ` `# option 1 +- ` ` ` `if` `canPutPatternHorizontally(rules,i,j,` `"+-"` `): ` ` ` `rules[i][j] ` `=` `"+"` ` ` `rules[i][j` `+` `1` `] ` `=` `"-"` ` ` ` ` `solveMagnets(rules,i,j) ` ` ` `# option 2 -+ ` ` ` ` ` `# option 3 xx ` ` ` `def` `checkConstraints(rules): ` ` ` ` ` `pCountH ` `=` `[` `0` `for` `i ` `in` `range` `(` `len` `(rules))] ` ` ` `nCountH ` `=` `[` `0` `for` `i ` `in` `range` `(` `len` `(rules))] ` ` ` `for` `row ` `in` `range` `(` `len` `(rules)): ` ` ` `for` `col ` `in` `range` `(` `len` `(rules[` `0` `])): ` ` ` `ch ` `=` `rules[row][col] ` ` ` `if` `ch ` `=` `=` `"+"` `: ` ` ` `pCountH[row] ` `+` `=` `1` ` ` `elif` `ch ` `=` `=` `"-"` `: ` ` ` `nCountH[row] ` `+` `=` `1` ` ` ` ` ` ` `pCountV ` `=` `[` `0` `for` `i ` `in` `range` `(` `len` `(rules[` `0` `]))] ` ` ` `nCountV ` `=` `[` `0` `for` `i ` `in` `range` `(` `len` `(rules[` `0` `]))] ` ` ` `for` `col ` `in` `range` `(` `len` `(rules[` `0` `])): ` ` ` `for` `row ` `in` `range` `(` `len` `(rules)): ` ` ` `ch ` `=` `rules[row][col] ` ` ` `if` `ch ` `=` `=` `"+"` `: ` ` ` `pCountV[col] ` `+` `=` `1` ` ` `elif` `ch ` `=` `=` `"-"` `: ` ` ` `nCountV[col] ` `+` `=` `1` ` ` ` ` ` ` `for` `row ` `in` `range` `(` `len` `(rules)): ` ` ` `if` `left[row] !` `=` `-` `1` `: ` ` ` `if` `pCountH[row] !` `=` `left[row]: ` ` ` `return` `False` ` ` `if` `right[row] !` `=` `-` `1` `: ` ` ` `if` `nCountH[row] !` `=` `right[row]: ` ` ` `return` `False` ` ` ` ` ` ` ` ` `for` `col ` `in` `range` `(` `len` `(rules[` `0` `])): ` ` ` `if` `top[col] !` `=` `-` `1` `: ` ` ` `if` `pCountV[col] !` `=` `top[col]: ` ` ` `return` `False` ` ` `if` `bottom[col] !` `=` `-` `1` `: ` ` ` `if` `nCountV[col] !` `=` `bottom[col]: ` ` ` `return` `False` ` ` `# ` ` ` `# if (top[col] != -1 and pCountH[col] != top[col]) or (bottom[col] != -1 and nCountH[col] != bottom[col]) : ` ` ` `# return False ` ` ` ` ` `return` `True` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `def` `solveMagnets(rules,i,j): ` ` ` ` ` `if` `i ` `=` `=` `len` `(rules) ` `and` `j ` `=` `=` `0` `: ` ` ` ` ` `# check the constraint before printing ` ` ` `if` `checkConstraints(rules): ` ` ` `print` `(rules) ` ` ` `elif` `j >` `=` `len` `(rules[` `0` `]): ` ` ` ` ` `solveMagnets(rules,i` `+` `1` `,` `0` `) ` ` ` ` ` `# normal cases ` ` ` `else` `: ` ` ` ` ` `if` `rules[i][j] ` `=` `=` `"L"` `: ` ` ` ` ` `# option 1 +- ` ` ` `if` `canPutPatternHorizontally(rules,i,j,` `"+-"` `): ` ` ` `rules[i][j] ` `=` `"+"` ` ` `rules[i][j` `+` `1` `] ` `=` `"-"` ` ` ` ` `solveMagnets(rules,i,j` `+` `2` `) ` ` ` ` ` `rules[i][j] ` `=` `"L"` ` ` `rules[i][j` `+` `1` `] ` `=` `"R"` ` ` ` ` `# option 2 -+ ` ` ` `if` `canPutPatternHorizontally(rules,i,j,` `"-+"` `): ` ` ` `rules[i][j] ` `=` `"-"` ` ` `rules[i][j` `+` `1` `] ` `=` `"+"` ` ` ` ` `solveMagnets(rules,i,j` `+` `2` `) ` ` ` ` ` `rules[i][j] ` `=` `"L"` ` ` `rules[i][j` `+` `1` `] ` `=` `"R"` ` ` ` ` `# option 3 xx ` ` ` `if` `True` `or` `canPutPatternHorizontally(rules,i,j,` `"xx"` `): ` ` ` `rules[i][j] ` `=` `"x"` ` ` `rules[i][j` `+` `1` `] ` `=` `"x"` ` ` ` ` `solveMagnets(rules,i,j` `+` `2` `) ` ` ` ` ` `rules[i][j] ` `=` `"L"` ` ` `rules[i][j` `+` `1` `] ` `=` `"R"` ` ` ` ` `# vertical check ` ` ` `elif` `rules[i][j] ` `=` `=` `"T"` `: ` ` ` ` ` `# option 1 +- ` ` ` `if` `canPutPatternVertically(rules,i,j,` `"+-"` `): ` ` ` `rules[i][j] ` `=` `"+"` ` ` `rules[i` `+` `1` `][j] ` `=` `"-"` ` ` ` ` `solveMagnets(rules,i,j` `+` `1` `) ` ` ` ` ` `rules[i][j] ` `=` `"T"` ` ` `rules[i` `+` `1` `][j] ` `=` `"B"` ` ` ` ` `# option 2 -+ ` ` ` `if` `canPutPatternVertically(rules,i,j,` `"-+"` `): ` ` ` `rules[i][j] ` `=` `"-"` ` ` `rules[i` `+` `1` `][j] ` `=` `"+"` ` ` ` ` `solveMagnets(rules,i,j` `+` `1` `) ` ` ` ` ` `rules[i][j] ` `=` `"T"` ` ` `rules[i` `+` `1` `][j] ` `=` `"B"` ` ` ` ` `# option 3 xx ` ` ` ` ` `if` `True` `or` `canPutPatternVertically(rules,i,j,` `"xx"` `): ` ` ` `rules[i][j] ` `=` `"x"` ` ` `rules[i` `+` `1` `][j] ` `=` `"x"` ` ` ` ` `solveMagnets(rules,i,j` `+` `1` `) ` ` ` ` ` `rules[i][j] ` `=` `"T"` ` ` `rules[i` `+` `1` `][j] ` `=` `"B"` ` ` ` ` `else` `: ` ` ` `solveMagnets(rules,i,j` `+` `1` `) ` ` ` ` ` `# Driver code ` `solveMagnets(rules,` `0` `,` `0` `) ` |

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

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