Which gives a tighter fit: a square peg in a round hole or a round peg in a square hole? (The tighter fit is the one for which the ratio of peg area to hole area is greater.)

### Solution

Consider the following picture:

Suppose that the side length of the large square is $a,$ so its area is $a^2.$ Then the circle has radius $a/2,$ so its area is $\pi a^2/4.$ By Pythagoras' theorem the small square has side length $\sqrt{a^2/2}$ so it's got area $a^2/2.$ The square peg in round hole fit is given by the ratio of the area of the small square to the area of the circle, which is $2/\pi.$ The round peg in square hole fit is given by the ratio of the area of the circle to the area of the large square, which is $\pi/4.$ Since $\pi > 3$ we have $\pi^2>8$ so $\pi > 8/\pi.$ Dividing by 4 gives $\pi/4 > 2/\pi$ so the round peg in a square hole gives the better fit.*This puzzle was contributed by Colm Mulcahy, Associate Professor of Mathematics at Spelman College in Atlanta, Georgia.
Colm's own puzzles have appeared in Math Horizons and in the New York Times. He is a long-time columnist for the Mathematical Association of America and has his own blog. You can follow Colm on Twitter.*