Billiards, Checkers, and Quadratic Reciprocity

Abstract

We indulge in what mathematicians call frivolous activities. In Arithmetic Billiards, a ball is bouncing around in a rectangle. In Parity Checkers we place checkers on a checkerboard under certain parity constraints. Both activities turn out to capture the division of congruence classes modulo a prime into squares and non-squares, allowing fairly simple proofs of the celebrated Law of Quadratic Reciprocity. Since the activities are analyzed somewhat in parallel we don't obtain two independent proofs. But Franz Lemmermeyer's online list of reciprocity proofs already contains well over three hundred items, which seems enough anyway.

Figures (6)

• Figure 1: Arithmetical billiards on a 5 by 7 rectangle. The bounces along the base occur at times 10, 20, and 30, which are congruent modulo 7 to $-4$, $+6$, and $+2$ respectively.
• Figure 2: Provided $m$ and $n$ are odd, the four bounces indicated will have the same sign.
• Figure 3: Each time the ball reaches the line $x=n-m$ from the left, it will make a tour through the rightmost $m$ by $m$ square and come back to the same point, continuing as if it had bounced against a wall at $x=n-m$.
• Figure 4: To solve the 6 by 10 checkers puzzle with a single pebble at square 6 of the bottom row, we two-color the path in 7 by 11 arithmetic billiards, changing color at the bounce at $(6,0)$. Then we place checkers on those dark squares of the checkerboard that correspond to two-colored crossings.
• Figure 5: When $m$ and $n$ have a common factor, we can find a nontrivial solution to the empty pebble-puzzle on an $m-1$ by $n-1$ checkerboard.

Theorems & Definitions (22)

• Lemma 1: Euler's criterion
• proof
• Theorem 2: Law of Quadratic Reciprocity
• Lemma 3
• Theorem 4
• proof
• Proposition 5
• proof
• Lemma 6
• proof