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

Famous quotes containing the words squares and/or square:

    “An afternoon of nurses and rumours;
    The provinces of his body revolted,
    The squares of his mind were empty,
    Silence invaded the suburbs,”
    —W.H. (Wystan Hugh)

    “Houses haunt me.
    That last house!
    How it sat like a square box!”
    —Anne Sexton (1928–1974)