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:
“And New York is the most beautiful city in the world? It is not far from it. No urban night is like the night there.... Squares after squares of flame, set up and cut into the aether. Here is our poetry, for we have pulled down the stars to our will.”
—Ezra Pound (1885–1972)
“Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.”
—Wallace Stevens (1879–1955)