🇺🇸 AMC 8 ⇄ switch contest
2014 AMC 8

Problem 23

Problem 23 · 2014 AMC 8 Hard
Logic & Word Problems logic-puzzleprimes-list

Three members of the Euclid Middle School girls' softball team had the following conversation.
Ashley: I just realized that our uniform numbers are all 2-digit primes.
Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month.
Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month.
Ashley: And the sum of your two uniform numbers is today's date.
What number does Caitlin wear?

Show answer
Answer: A — 11.
Show hints
Hint 1 of 3
The dates do double duty as ordering clues: every pairwise sum is a calendar date (1–31), and the words "earlier" / "later" / "today" rank those three sums smallest < middle < largest.
Still stuck? Show hint 2 →
Hint 2 of 3
Sums ≤ 31 squeeze the three primes down to a tiny list — find a triple of two-digit primes whose three pairwise sums all fit and are distinct.
Still stuck? Show hint 3 →
Hint 3 of 3
Each girl names the sum of the OTHER two. So the girl with the largest sum-of-others must herself be the smallest number — the big two are added without her.
Show solution
Approach: shrink the prime list, then use date order to rank the sums
  1. Each pairwise sum is a date ≤ 31. Two-digit primes are 11, 13, 17, 19, 23, 29, but any pair using 23 or 29 overshoots 31, so the three numbers come from {11, 13, 17, 19}.
  2. Pick a triple with three distinct sums: {11, 13, 17} gives 11+13 = 24, 11+17 = 28, 13+17 = 30 — all distinct and ≤ 31. ✓
  3. Order the clues: Bethany's date (earlier) is the smallest sum 24, today is the middle 28, Caitlin's date (later) is the largest 30. Each girl quotes the sum of the OTHER two.
  4. Caitlin's date = 30 = sum of the other two = Ashley + Bethany, so Ashley and Bethany are the two larger numbers {13, 17}, leaving Caitlin = 11.
  5. Cross-check: Bethany's date 24 = Ashley + Caitlin = Ashley + 11 ⇒ Ashley = 13, so Bethany = 17, and today = Ashley + Bethany... wait, today = Bethany + Caitlin = 17 + 11 = 28. ✓ All three statements hold.
  6. Caitlin wears 11.
  7. Why this transfers: when each clue references "the others," the person tied to the biggest total is the one left out of it — so largest sum-of-others ↔ smallest own value. Spotting that inverse ordering cracks many "sum of the other two" puzzles.
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