Largest Magic Square
<p>A <code>k x k</code> <strong>magic square</strong> is a <code>k x k</code> grid filled with integers such that every row sum, every column sum, and both diagonal sums are <strong>all equal</strong>. The integers in the magic square <strong>do not have to be distinct</strong>. Every <code>1 x 1</code> grid is trivially a <strong>magic square</strong>.</p>

<p>Given an <code>m x n</code> integer <code>grid</code>, return <em>the <strong>size</strong> (i.e., the side length </em><code>k</code><em>) of the <strong>largest magic square</strong> that can be found within this grid</em>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1800-1899/1895.Largest%20Magic%20Square/images/magicsquare-grid.jpg" style="width: 413px; height: 335px;" />
<pre>
<strong>Input:</strong> grid = [[7,1,4,5,6],[2,5,1,6,4],[1,5,4,3,2],[1,2,7,3,4]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The largest magic square has a size of 3.
Every row sum, column sum, and diagonal sum of this magic square is equal to 12.
- Row sums: 5+1+6 = 5+4+3 = 2+7+3 = 12
- Column sums: 5+5+2 = 1+4+7 = 6+3+3 = 12
- Diagonal sums: 5+4+3 = 6+4+2 = 12
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://fastly.jsdelivr.net/gh/doocs/leetcode@main/solution/1800-1899/1895.Largest%20Magic%20Square/images/magicsquare2-grid.jpg" style="width: 333px; height: 255px;" />
<pre>
<strong>Input:</strong> grid = [[5,1,3,1],[9,3,3,1],[1,3,3,8]]
<strong>Output:</strong> 2
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 50</code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 10<sup>6</sup></code></li>
</ul>
Array; Matrix; Prefix Sum
function largestMagicSquare(grid: number[][]): number {
    let m = grid.length,
        n = grid[0].length;
    // 前缀和
    let rowSum = Array.from({ length: m + 1 }, (v, i) => new Array(n + 1).fill(0)),
        colSum = Array.from({ length: m + 1 }, v => new Array(n + 1).fill(0));
    for (let i = 0; i < m; i++) {
        rowSum[i + 1][1] = grid[i][0];
        for (let j = 1; j < n; j++) {
            rowSum[i + 1][j + 1] = rowSum[i + 1][j] + grid[i][j];
        }
    }

    for (let j = 0; j < n; j++) {
        colSum[1][j + 1] = grid[0][j];
        for (let i = 1; i < m; i++) {
            colSum[i + 1][j + 1] = colSum[i][j + 1] + grid[i][j];
        }
    }
    // console.log(rowSum, colSum)
    // 寻找最大k
    for (let k = Math.min(m, n); k > 1; k--) {
        for (let i = 0; i + k - 1 < m; i++) {
            for (let j = 0; j + k - 1 < n; j++) {
                let x2 = i + k - 1,
                    y2 = j + k - 1;
                if (valid(grid, rowSum, colSum, i, j, x2, y2)) {
                    return k;
                }
            }
        }
    }
    return 1;
}

function valid(
    grid: number[][],
    rowSum: number[][],
    colSum: number[][],
    x1: number,
    y1: number,
    x2: number,
    y2: number,
): boolean {
    let diff = rowSum[x1 + 1][y2 + 1] - rowSum[x1 + 1][y1];
    // 行
    for (let i = x1 + 1; i <= x2; i++) {
        if (diff != rowSum[i + 1][y2 + 1] - rowSum[i + 1][y1]) {
            return false;
        }
    }
    // 列
    for (let j = y1; j <= y2; j++) {
        if (diff != colSum[x2 + 1][j + 1] - colSum[x1][j + 1]) {
            return false;
        }
    }
    // 主队对角线
    let mainSum = diff;
    for (let i = x1, j = y1; i <= x2; i++, j++) {
        mainSum -= grid[i][j];
    }
    if (mainSum != 0) return false;
    // 副对角线
    let subSum = diff;
    for (let i = x1, j = y2; i <= x2; i++, j--) {
        subSum -= grid[i][j];
    }
    if (subSum != 0) return false;
    return true;
}
