AMC 10 · 2013 · #7
Grade 4 countingPick an answer.
The counting has a clean repeating rule: turn n contributes exactly n numbers. Spot that pattern, add the turn sizes to see how many numbers have been said by the end of each turn, then find which turn the 53rd number falls in and read off its value inside that turn.
Count numbers per turn
Turn number n says exactly that many numbers.
The turn number is the same as how many numbers get said that turn.
The turn number is the same as how many numbers get said on that turn.
▸ Why?
Each turn says one more number than the turn before, so the counts climb by the same fixed step.
▸ Why?
Every number said is matched with exactly one position in its turn, so counting positions counts the numbers.
Add up to find the turn
Adding turns pins the target to turn ten.
Keep a running total of numbers said until it first reaches past 53.
4.NBT.B.4Identify SubproblemsLocate 53 inside turn 10
Its place inside that turn gives 8, choice (B).
Subtract the numbers already counted, then step that far into the current turn's list.
4.OA.A.3Make A Systematic ListAdd up how many numbers each turn says until you pass the one you want, then count that many steps into the last turn's list.
- Count numbers per turn
- Add up to find the turn
- Locate 53 inside turn 10