AMC 10 · 2023 · #5

Grade 4 geometry-2d
Archetype Casework by Position / Vertex Type └ Sub-archetype Grid / Solid Cell Position
parity-coloringspatial-visualizationsystematic-enumerationcomplementary-counting parity-coloringcomplementary-countingcasework ↑ Prerequisites: parity-coloringspatial-visualization
📏 Medium solution 💡 3 insights
📘 View easy version →
Problem
A hidden domino covers two side-by-side cells of a three by three grid. Each turn you name a cell and are told whether it is covered. Find the smallest number of turns that guarantees naming at least one covered cell, wherever the domino lies.

Pick an answer.

(A)
3
(B)
5
(C)
4
(D)
8
(E)
6
How to solve
Strategy Draw a Diagram

Tool #1 (Draw a Diagram) leads — color the 3 × 3 grid like a checkerboard with corners white. There are 5 white and 4 black cells, and because the domino covers two adjacent cells it always covers one white and one black. That single observation unlocks the upper bound: name all 4 black cells and you must hit one of the domino's two cells. Tool #16 (Change Focus / Complement) gives the matching lower bound — instead of asking "how few cells let me hit?" ask "how many cells can I leave un-named without leaving a domino-shaped gap?". The un-named cells must contain no adjacent pair, and the largest such set on a 3 × 3 grid is the 5-cell white set. So at most 5 cells can be un-named, meaning at least 9 - 5 = 4 must be named. Tool #6 (Guess and Check) acts as a sanity sweep across small strategies. Algebra is unnecessary — the picture and the complement count are decisive.

1STEP 1

Colour it in two colours

Colour the grid in two colours.

W & B & W ; B & W & B ; W & B & W ; counts: 5 white, 4 black
2STEP 2

What a domino covers

A domino covers one of each colour.

domino = (one white cell) + (one black cell)
3STEP 3

Name every cell of the smaller colour

Naming all of the smaller colour guarantees a hit.

Name all 4 black cells → guaranteed hit
4STEP 4

Why fewer fails

With fewer, a domino can hide.

un-named ≤ 5 → named ≥ 9 - 5 = 4
5STEP 5

Read the minimum

The minimum is 4.

minimum turns = 4 → (C)
Answer
4
Three checks. (1) Strategy sanity: writing out all twelve possible domino positions (6 horizontal + 6 vertical) and checking that each one covers at least one black cell confirms the upper-bound strategy actually works. (2) Lower-bound sanity: try 3 guesses at corners, say (1,1), (1,3), (3,1). The un-named set is {(1,2), (2,1), (2,2), (2,3), (3,2), (3,3)} — and the pair (2,2), (2,3) is adjacent, so a horizontal domino fits there with no hit. So 3 really is not enough. (3) Eliminate distractors: (A) 3 is the most natural too-low guess (refuted above); (B) 5 is the white-set strategy, which works but uses more turns than needed; (D) 8 and (E) 6 are over-counts that ignore the parity insight.
💡Key takeaway

This AMC 12 problem only needs Grade 4 grid patterns you already know — color the 3 × 3 board like a checkerboard (5 white, 4 black), notice every 2 × 1 domino must cover one of each, so naming the 4 black cells always guarantees a hit and 3 guesses can never be enough.

More like this

Same archetype — closest grade level first.