# 2008 iTest Problems/Problem 12

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## Problem

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where a and b are relatively prime positive integers. Find $a+b$.

## Solution

The successful outcomes are easy to count by hand. Below is a table (each row represents Tony’s outcome and each column represents Mitchell’s outcome) where dots signify a successful result.

 1 2 3 4 5 6 1 $\cdot$ 2 $\cdot$ 3 $\cdot$ 4 $\cdot$ 5 $\cdot$ 6 $\cdot$

The number of successful outcomes is $6$, and the total number of possibilities is $36$, so the probability that the positive difference of the two dice rolls is three is $\tfrac{1}{6}$. Thus, $a+b=\boxed{7}$.