Problem 12 · 2022 AMC 8
Medium
Counting & Probability
careful-countingperfect-square
Show answer
Answer: B — 1/8.
Show hints
Hint 1 of 2
“10 times A plus B” is just sticking A in the tens place and B in the ones place — so N is a 2-digit number. Don't list spinner combos; list the perfect squares that could appear, since there are very few.
Still stuck? Show hint 2 →
Hint 2 of 2
Spinner A ∈ {5, 6, 7, 8} puts N in the 50s–80s, and B ∈ {1, 2, 3, 4} fixes the ones digit. Which perfect squares live in that window?
Show solution
Approach: count the rare outcomes (perfect squares), not the common ones
- Insight: N = 10A + B just glues the two spinner numbers into a 2-digit number (A tens, B ones). Rather than checking all 16 spins, hunt the scarce target: perfect squares are rare, so list them.
- With A ∈ {5,6,7,8} and B ∈ {1,2,3,4}, N runs 51 to 84. The only perfect squares there are 64 = 82 and 81 = 92. Check each is reachable: 64 needs A=6, B=4 ✓; 81 needs A=8, B=1 ✓.
- So 2 of the 4 × 4 = 16 equally likely outcomes win: probability = 216 = 18.
- You'll see this again: when favorable outcomes are rare, count them directly instead of sifting the whole sample space — far fewer cases to check.
Mark:
· log in to save