In PHP, what strategies can be employed to count the number of possible solutions for a Sudoku field while avoiding redundant computations?

To count the number of possible solutions for a Sudoku field without redundant computations, we can use backtracking algorithm. This algorithm involves trying out different possible values for each empty cell in the Sudoku grid and backtracking if a solution is not valid. By keeping track of the number of valid solutions found, we can avoid redundant computations. 

```php
<?php
function countSudokuSolutions($board) {
    return solveSudoku($board);
}

function solveSudoku(&$board) {
    $emptyCell = findEmptyCell($board);
    
    if($emptyCell === null) {
        return 1; // found a valid solution
    }
    
    $row = $emptyCell[0];
    $col = $emptyCell[1];
    
    $count = 0;
    
    for($num = 1; $num <= 9; $num++) {
        if(isValidMove($board, $row, $col, $num)) {
            $board[$row][$col] = $num;
            
            $count += solveSudoku($board);
            
            $board[$row][$col] = 0; // backtrack
        }
    }
    
    return $count;
}

function findEmptyCell($board) {
    for($i = 0; $i < 9; $i++) {
        for($j = 0; $j < 9; $j++) {
            if($board[$i][$j] === 0) {
                return [$i, $j];
            }
        }
    }
    
    return null;
}

function isValidMove($board, $row, $col, $num) {
    // Check row
    for($i = 0; $i < 9; $i++) {
        if($board[$row][$i] === $num) {
            return false;
        }
    }
    
    // Check column
    for($i = 0; $i < 9; $i++) {
        if($board[$i][$col] === $num) {
            return false;
        }
    }
    
    // Check 3x3 subgrid
    $startRow = $row - $row % 3;
    $startCol = $col - $col % 3;
    
    for($i = 0; $i < 3; $i++) {
        for($j = 0; $

Keywords

PHP Sudoku solutions counting redundant computations

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