jmc

Question

Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains 8 unit squares. The second ring contains 16 unit squares. If we continue this process, then what is the number of unit squares in the

10 0^{t h}

ring?

Accepted Answer

The n^th ring can be partitioned into four rectangles: two containing 2n+1 unit squares and two containing 2n-1 unit squares. So there are 2(2n+1)+2(2n-1) = 8n unit squares in the n^th ring. Thus, the 100^th ring has 8 100 = 800 unit squares.