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 (19281974)
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