Rotate Image

A 90-degree clockwise rotation maps position (i,j) to (j, n-1-i). This is two simpler operations composed: transpose across the diagonal, then reverse each row. Neither needs extra memory beyond a swap variable.

Full solution

Transpose across the main diagonal, then reverse each row: two loops anyone can derive on the spot, O(n^2) time with O(1) extra space, and no four-way layer swap whose bounds are easy to get wrong.

from typing import List


def rotate(matrix: List[List[int]]) -> None:
    n = len(matrix)
    # transpose: swap only above the diagonal, or every pair is swapped back
    for i in range(n):
        for j in range(i + 1, n):
            matrix[i][j], matrix[j][i] = matrix[j][i], matrix[i][j]
    # reverse each row: transpose alone is a reflection, not a rotation
    for row in matrix:
        row.reverse()

function rotate(matrix) {
  const n = matrix.length;
  // transpose: swap only above the diagonal, or every pair is swapped back
  for (let i = 0; i < n; i++) {
    for (let j = i + 1; j < n; j++) {
      [matrix[i][j], matrix[j][i]] = [matrix[j][i], matrix[i][j]];
    }
  }
  // reverse each row: transpose alone is a reflection, not a rotation
  for (const row of matrix) row.reverse();
}

The theory behind it

Matrix — the ground this problem stands on. All Matrix problems

What Matrix is

A matrix is a flat grid of cells laid out in rows and columns, like a checkerboard or a spreadsheet. Each cell has an address made of two numbers: its row index going down and its column index going across. Because computer memory stores these rows one after another or as arrays of arrays, jumping straight to any individual cell takes the same tiny fraction of time no matter where it sits on the board.

When to reach for it

Reach for matrix techniques when the input is a two-dimensional grid, board, or map. Common prompts include walking through a maze, finding connected islands of land in water, rotating an image ninety degrees in place, or reading numbers in a spiral. Words like rows, columns, neighbors, adjacent squares, or diagonal lines are clear signs. It also appears in grid planning where each cell depends on the values above and to the left.

How the pattern works

Think of navigating the grid using directional coordinate offsets. Define row and column step arrays for moving up, down, left, and right. Always check that new row and column indices stay strictly between zero and the board boundaries before reading cell values. For spiral scans, track four boundary lines for top, bottom, left, and right, shrinking them inward after reading each side. When searching connected regions, mark visited cells directly or write a marker value into the cell to avoid looping.