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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2021 JMPSC Sprint Problems/Problem 19

Problem

Solution