I am currently doing a unit on combinatorics (the mathematical study of counting) with my precalculus students, and I was inspired to post a few counting-themed challenge problems for your enjoyment. (Also, it’s my spring break!)
As you probably know, a chess board consists of 64 squares arranged in eight rows and eight columns.
- How many squares—of any size—are on a chess board? Each of the 64 smallest squares count, of course, but there are also larger ones; for example, the four squares in the upper left corner form a 2×2 square.
- How many squares of any size would there be on a 9 by 9 chess board? 10 by 10? n by n?
- How many rectangles of any size and shape are there on a chess board? Is this easier or harder than counting squares?
- How about rectangles on 9 by 9, 10 by 10, n by n, or m by n chess board?