AMC 10 · 2011 · #23

Grade 6 arithmetic
Archetype Modular Cycle Frequency Count └ Sub-archetype Position-in-Cycle Lookup
base-conversionsequences-geometric pattern-recognitionconvert-to-algebra ↑ Prerequisites: pattern-recognition
📏 Medium solution 💡 3 insights
Problem
Seven students, Alice through George in alphabetical order, take turns dealing with the integers from 1 to 1000. On her turn, a student considers every number no earlier student has said, groups those numbers in consecutive threes, and skips the middle number of each group while saying the rest. George ends up saying exactly one number that nobody else says. Which number is it?

Pick an answer.

(A)
37
(B)
242
(C)
365
(D)
728
(E)
998

AMC 10 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Chasing 1000 numbers through seven students by hand is hopeless, so the move is Tool #5 (Look for a Pattern): watch how one student's leftovers turn into the next student's list. Tool #4 (Introduce a Variable) captures that hand-off cleanly — a list with first term a and common difference d passes on a list with first term a+d and difference 3d. Tool #7 (Identify Subproblems) splits the work into two easy tracks: how the difference grows (powers of three) and how the starting number grows (one addition per student). Tool #3 (Eliminate Possibilities) finishes: George's very next number already passes 1000, so only one survives.

1STEP 1

What Alice leaves behind

Alice skips the middle of every triple: 2, 5, 8, …, 998. That skipped list is exactly Barbara's — gap 3.

2, 5, 8, …, 998 — first term 2, common difference 3
2STEP 2

The hand-off rule, with variables

A list with first term a and gap d skips a+d, a+4d, a+7d, … — a list with first term a+d, gap 3d.

(a, d) ⟶ (a+d, 3d)
3STEP 3

The gap triples each round

Alice's gap is 1 and every hand-off triples it, so the seven gaps are 1, 3, 9, 27, 81, 243, 729.

d₁, …, d₇ = 3⁰, 3¹, …, 3⁶ = 1, 3, 9, 27, 81, 243, 729
4STEP 4

Track each starting number

Each new start is the old start plus the old gap: 1, 2, 5, 14, 41, 122 — so George starts at 365.

a₁,…,a₇ = 1, 2, 5, 14, 41, 122, 365
5STEP 5

George is left with one number

George's next number 365 + 729 = 1094 is past 1000, so he says only 365 — choice (C).

365, 1094, …; only 365 ≤ 1000 → (C) 365
Answer
365
A closed form confirms the last start: a₇ = 1 + (1 + 3 + 9 + 27 + 81 + 243) = 1 + 364 = 365, equivalently 1 + (3⁶ - 1)/2 = 1 + 364 = 365. It also makes sense that only George gets a lone number: the earlier gaps 1, 3, 9, 27, 81, 243 all fit inside 1000 many times, so those students say lots of numbers, but George's gap of 729 leaves room for just one value below 1000. Finally, 365 is exactly answer (C), while the distractors 37, 242, 728, 998 never appear as a starting number in the sequence.
💡Key takeaway

Whatever numbers one student skips become the next student's list — and each hand-off triples the gap (the powers of three) while nudging the start up by one gap. Follow that for seven students and George is left with just 365.

  • What Alice leaves behind
  • The hand-off rule, with variables
  • The gap triples each round
  • Track each starting number
  • George is left with one number

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