# 2021 JMPSC Sprint Problems/Problem 19

## Problem

As an April Fool’s prank, Sean hacks his teacher’s digital clock and switches each digit to a certain letter. Right now, the hacked clock displays $\textbf{M:AT}$. $14$ minutes later, it displays $\textbf{A:TM}$. If no two digits represent the same letter, find the value of $\textbf{A} \times \textbf{M}.$

## Solution

Note that if the $M$ changes to $A$ in just 14 minutes, $M:AT$ is at the end of an hour, meaning that $A$ is either $4$ or $5$. If $A=4$, T must be $0$ if $14$ minutes later is a new hour. However, at time $14$ minutes after $M:40$ will be $M:54$ instead of a new hour number. Thus, $A=5$. If in $14$ minutes it is due to be $5:TM$, then it is between 4-o'clock and 5-o 'clock currently. Thus, from $M:AT$, $M=4$. From $5:T4$ ( $A:TM$), we can guess that $T=0$. Then we have $M:AT=4:50$ and $A:TM=5:04$. Checking, the times are indeed 14 minutes apart! We want $A \times M$, so the answer is $5 \times 4 = \boxed{20}$.

~MathHayden

(wow this is my *one* actual contribution to the AoPS wiki!)

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