2025/26 AHSMC Part 1 Problem 4

Problem 4 of 16

A rectangular pool table is placed on a grid and has corners at $(0,0)$ , $(5,0)$ , $(0,3)$ , and $(5,3)$ . There are pockets only in the four corners. A ball is hit from the corner at $(0,0)$ along the line $y = x$ . How many times does the ball bounce off a wall before entering a pocket?

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Solution

Unfold the table into a grid of $5\times 3$ rectangles and travel in a straight line along $y=x$ . The first lattice point $(x,y)$ with $x$ a multiple of $5$ , $y$ a multiple of $3$ , and $y=x$ is $(15,15)$ . Vertical wall-crossings (excluding the pocket) number $\dfrac{15}{5}-1 = 2$ ; horizontal wall-crossings number $\dfrac{15}{3}-1 = 4$ . Total bounces: $2+4 = 6$ . The answer is (C).

University of Alberta

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