AMC 8 · 2007 · #11

Grade 4 logic

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

logical-deductionsystematic-enumeration caseworksystematic-enumeration ↑ Prerequisites: logical-deduction

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Four square tiles I, II, III, IV each have a number on each of their four sides. They are slid (without rotation or reflection) into a 2 × 2 grid of rectangles A (top-left), B (top-right), C (bottom-left), D (bottom-right). Wherever two tiles touch, the two numbers on the shared edge must match. Which tile lands in rectangle C?

Pick an answer.

(A)

(B)

(C)

III

(D)

(E)

cannot be determined

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

Tool #2 (Make a Systematic List) lines up all 16 tile-edge numbers in a table so nothing is missed. Tool #13 (Count Systematically) then tallies how often each value 0-9 appears. The key observation: any number on an interior edge of the 2 × 2 grid is shared by two tiles, so its value must appear at least twice across the four tiles. A value that appears only once cannot lie on an interior edge — it must be on the outer boundary. That uniqueness forces one tile into a specific corner, and the remaining three are pinned down by matching the shared edges.

1STEP 1

List every edge value for every tile in a table.

& top & right & bottom & left ; I & 8 & 9 & 7 & 3 ; II & 6 & 3 & 2 & 4 ; III & 7 & 5 & 0 & 1 ; IV & 2 & 1 & 6 & 9

2STEP 2

Tally each digit over all 16 edges: only 0 and 5 appear exactly once, both on tile III (0 bottom, 5 right).

Frequency: 0{:}1, 1{:}2, 2{:}2, 3{:}2, 4{:}1, 5{:}1, 6{:}2, 7{:}2, 8{:}1, 9{:}2

3STEP 3

Unique digits can't sit on an interior edge, so III's 0 and 5 face out — only the bottom-right corner does that, so III goes in D.

III → D

4STEP 4

III's top 7 needs a tile with bottom 7 above it (tile I → B), and its left 1 needs a tile with right 1 beside it (tile IV → C).

I → B, IV → C

5STEP 5

Only tile II remains, so II → A; its shared edges match too (3 = 3 with B, 2 = 2 with C).

II → A; 3 = 3, 2 = 2 ✓

6STEP 6

Read off the answer.

C contains tile IV → (D)

Answer

Double-check every shared edge in the final layout smallmatrix II & I ; IV & III smallmatrix: A-B (II's right 3 = I's left 3) ✓; C-D (IV's right 1 = III's left 1) ✓; A-C (II's bottom 2 = IV's top 2) ✓; B-D (I's bottom 7 = III's top 7) ✓. All four interior edges match. The unique values 0, 4, 5, 8 all end up on the outer boundary, exactly as the uniqueness argument predicted.

💡Key takeaway

Tally the digits on every tile edge. The ones that show up only once cannot match anything, so they must face the outside — and that single observation pins the puzzle down.

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