Problem 24 · 2008 AMC 8
Hard
Counting & Probability
case-on-die-roll
Ten tiles numbered 1 through 10 are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
Show answer
Answer: C — 11/60.
Show hints
Hint 1 of 2
The die only shows 1–6, so loop over its six values — fewer cases than looping over the ten tiles.
Still stuck? Show hint 2 →
Hint 2 of 2
For each die value d, ask which tiles make d·t a perfect square: the tile must supply exactly the prime factors d is missing to make every exponent even.
Show solution
Approach: case on the die value (only six cases)
- Total outcomes: 10 tiles × 6 die faces = 60. Now scan the die. d = 1 is already a square, so any square tile works: t = 1, 4, 9 ⇒ 3.
- d = 2 needs the tile to contribute another factor of 2 (and otherwise be square): t = 2, 8 ⇒ 2. d = 3 needs another 3: only t = 3 ⇒ 1.
- d = 4 is itself a square, so again the square tiles: t = 1, 4, 9 ⇒ 3. d = 5: only t = 5 ⇒ 1. d = 6: only t = 6 ⇒ 1.
- Successes: 3 + 2 + 1 + 3 + 1 + 1 = 11, so the probability is 11/60.
- Why this transfers: a number is a perfect square exactly when every prime appears an even number of times — so "make this product square" means the other factor must fill in the odd-power primes.
Mark:
· log in to save