AMC 10 · 2016 · #13

Grade 7 arithmetic
Archetype Pair-Sum Invariant └ Sub-archetype Universe Total Minus Known
logical-deductionpermutations-basic invariant-monovariantcasework ↑ Prerequisites: logical-deduction
📏 Medium solution 💡 3 insights
Problem
Five friends fill seats 1 through 5 in a row, left to right. While Ada is away, Bea moves 2 seats to the right, Ceci moves 1 seat to the left, and Dee and Edie swap seats. After everyone shifts, the only open seat for Ada is one of the two end seats. Find the seat Ada was sitting in before she left.

Pick an answer.

(A)
1
(B)
2
(C)
3
(D)
4
(E)
5

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Instead of hunting for the whole seating chart, tool #4 (Introduce a Variable) measures each person by a single signed number: how far they moved, with right as + and left as -. The key fact is that nobody enters or leaves the row, so every seat someone leaves is taken by someone else, which forces all five signed moves to add up to 0. Tool #1 (Draw a Diagram) keeps the left-right number line straight in mind, and once Ada's move is known, tool #3 (Eliminate Possibilities) uses the end-seat clue to rule out one of the two ends and pin down where she started.

1STEP 1

Turn each move into a signed number

Take right as +, left as -, so Bea is +2, Ceci is -1, and the Dee–Edie swap is 0.

Bea=+2, Ceci=-1, Dee+Edie=0
2STEP 2

All five moves add to zero

The same five seats are refilled, just rearranged, so all five signed moves must add to 0; call Ada's move a.

2 + (-1) + 0 + a = 0
3STEP 3

Solve for Ada's move

Adding the known moves gives 2 + (-1) + 0 = 1, so Ada's move must cancel it: a = -1, one seat left.

1 + a = 0 → a = -1
4STEP 4

Use the end-seat clue to find the start

A left step can't reach seat 5 (that needs seat 6), so Ada landed on seat 1 and started one seat right, at seat 2 — (B).

end ∈ {1,5}, moved -1 → end=1, start=1-(-1)=2 → (B)
Answer
2
Build a chart that fits and check it. Start with Ada 2, Bea 1, Ceci 3, Dee 4, Edie 5. Then Bea 1→3, Ceci 3→2, Dee and Edie swap to 5 and 4. The taken seats become {2,3,4,5}, leaving seat 1 open for Ada, an end seat. Every rule holds and Ada started in seat 2, confirming (B).
💡Key takeaway

Tag each move + for right and - for left; since no one leaves the row, all the moves cancel to zero, so Ada's must be -1, putting her start at seat 2.

  • Turn each move into a signed number
  • All five moves add to zero
  • Solve for Ada's move
  • Use the end-seat clue to find the start

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