Square Pyramidal Number - Squares in A Square

Squares in A Square

A common mathematical puzzle involves finding the number of squares in a large n by n square grid. This number can be derived as follows:

  • The number of 1×1 boxes found in the grid is .
  • The number of 2×2 boxes found in the grid is . These can be counted by counting all of the possible upper-left corners of 2×2 boxes.
  • The number of k×k boxes (1 ≤ k ≤ n) found in the grid is . These can be counted by counting all of the possible upper-left corners of k×k boxes.

It follows that the number of squares in an n by n square grid is:

That is, the solution to the puzzle is given by the square pyramidal numbers.

The number of rectangles in a square grid is given by the squared triangular numbers.

Read more about this topic:  Square Pyramidal Number

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