# 2021 JMPSC Sprint Problems/Problem 14

## Problem

Ari, Bryant, Chandler, and David each tell one truth and one lie.

1. Ari: Bryant is the tallest among the four of us. Chandler is the shortest among the four of us.
2. Bryant: Ari is the oldest in the room. Ari is the shortest in the room.
3. Chandler: David is taller than me. David is older than me.
4. David: Chandler told two truths. I am the oldest person in the room.

If the first name of the shortest person is $a$ letters long and the first name of the second-tallest person is $b$ letters long, find $a \times b.$ (Assume that no two people share the same height and are born on the same day.)

## Solution

Chandler did not tell two truths, so David's first statement is false. So David is the oldest person in the room.

Chandler's second statement is true because David is the oldest, so his first statement is a lie. Therefore, Chandler is taller than David.

Bryant's first statement is a lie because Ari isn't the oldest in the room. So Ari is the shortest.

Since Chandler is taller than David, Chandler is not the shortest among the four of them. Therefore, Bryant is the tallest.

Ari, David, Chandler, Bryant is the ordering of them from shortest to tallest. So Ari( $3$ letters) is the shortest in the room and Chandler( $8$ letters) is the second tallest in the room. $a=3, b=8.\ 3\cdot8=\boxed{24}$.

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