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!)

See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png

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