Competition · AMC preparation · step 4 of 4

AMC 8 · 2025 · #11

Grade 3 geometry-2dcounting

Archetype Grid Tiling / Packing with Divisibility └ Sub-archetype Fixed Piece Set — Feasibility by Elimination

area-rectanglesparity-coloringspatial-visualizationparity parity-coloringcaseworksystematic-enumeration ↑ Prerequisites: area-rectanglesparity

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Problem

A 3 × 4 rectangle is completely covered by three tetrominoes chosen from the five shapes I, O, L, T, S (rotations and reflections allowed). One of the three tiles must be an S tile. Which pair of tiles makes up the other two?

Pick an answer.

(A)

I and L

(B)

I and T

(C)

L and L

(D)

L and S

(E)

O and T

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tiling problems live on a picture, so Tool #1 (Draw a Diagram) is the starting move: color the 3 × 4 board like a chessboard and count how many black/white squares each tetromino must cover. That single picture turns a hard geometry question into easy parity arithmetic. Tool #3 (Eliminate Possibilities) then knocks out the choices that violate the black/white balance — this is the classic AMC multiple-choice move. Finally, Tool #10 (Physical Representation) finishes the job: with only three options left, cut out paper tetrominoes (or shade cells on graph paper) and actually try to place an S together with the candidates. The combination that fits is the answer.

1STEP 1

Color the board like a chessboard

Color the 3 × 4 board like a chessboard: the 12 squares split into 6 black and 6 white.

3 × 4 = 12, 12 ÷ 2 = 6 black + 6 white

2STEP 2

Count each tile's colors

On a chessboard, an I, O, L, or S tile covers 2 black + 2 white, but a T tile covers 3 of one color and 1 of the other.

I, O, L, S → 2B+2W ; T → 3B+1W or 1B+3W

3STEP 3

Rule out choices by color

The S tile takes 2 black + 2 white, leaving 4 black and 4 white; a T pairs only with another T — eliminate (B) and (E).

Remaining after S: 6-2=4 B, 6-2=4 W

Checking that 'black count = white count' is even-vs-odd / balance reasoning — a Grade 2 odd-or-even style argument.

A T tetromino can be part of the tiling only if a second T tetromino is used with it, so the two partners of the mandatory S tile cannot be one T tile together with a non-T tile.

▸ Why?

Color the board like a chessboard: it then holds exactly 6 black and 6 white squares, and because the three tiles cover it with no gaps or overlaps, the black cells of the tiles must total 6 and the white cells must total 6.

▸ Why?

The 3x4 board has 12 cells, and in every row of four the chessboard coloring alternates to give 2 black and 2 white, so the whole board pairs each black square with a white one into 6 black and 6 white.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

The three tiles sit in the board with no gaps or overlaps, so every black board cell belongs to exactly one tile, and the tiles' black cells added together must equal the board's 6 black cells.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

Each I, O, L, or S tile always covers 2 black and 2 white, but a T tile always covers 3 of one color and 1 of the other; so after the S takes 2 black and 2 white, the leftover 4 black and 4 white can absorb one T's lopsided 3-and-1 only if another T supplies the matching 1-and-3.

▸ Why?

A tetromino is four squares joined edge to edge, and on a chessboard two edge-joined squares are always opposite colors, so how the four cells sit fixes each shape's split — 2 and 2 for I, O, L, S, and 3 and 1 for the T's cross shape.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

The mandatory S covers 2 black and 2 white, so removing it from the board's 6 black and 6 white leaves exactly 4 black and 4 white for the other two tiles.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The last two tiles must cover 4 black cells altogether, and 4 is even; each I, O, L, or S covers 2 black cells — an even count — while a T covers 3 or 1, an odd count. An odd count and an even count can never add up to an even total, so a T's odd contribution can only be completed to the even total of 4 by another odd contribution, and among the five shapes only a second T supplies one.

Even and odd (parity)A whole number is even when it splits into two equal whole groups (a factor of 2), else odd; odd×odd stays odd and even×anything is even.

2.OA.C.3Eliminate Possibilities

4STEP 4

Test the remaining choices

(A)'s I-tile forces a 2 × 4 strip that can't hold an S, and (D)'s S + S + L always leaves a gap, so both (A) and (D) fail.

(A) fails: I forces a full row → 2 × 4 strip cannot be L + S

5STEP 5

Place the winning tiles

Placing the S tile bottom-left splits the eight empty cells into two L-shaped regions — a real S + L + L tiling exists, giving (C).

S at {(1,2),(2,2),(2,1),(3,1)} + L at {(1,1),(1,3),(2,3),(3,3)} + L at {(3,2),(4,1),(4,2),(4,3)} → (C)

Answer

L and L

Three tiles of 4 squares cover 12 squares, which matches the 3 × 4 = 12 area of the board — the area arithmetic checks out. The chessboard parity (6 black, 6 white) is preserved by the S tile (2+2) and by the two L tiles (2+2 each), summing to 6+6. The explicit placement of S, L, L given above tiles the rectangle with no overlaps and no holes (one can verify by shading the cells on graph paper), so the answer (C) L and L is both internally consistent and constructively demonstrated.

💡Key takeaway

This AMC 8 problem only needs Grade 3 unit-square counting (and a clever chessboard picture) that you already know!

Color the board like a chessboard
Count each tile's colors
Rule out choices by color
Test the remaining choices
Place the winning tiles

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Color the board like a chessboard

Count each tile's colors

Rule out choices by color

Test the remaining choices

Place the winning tiles

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