AMC 8 · 2002 · #23

Grade 4 geometry-2dcounting

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Periodic Placement Formula

pattern-recognitionspatial-visualizationfraction-arithmeticratio-proportion pattern-recognitionidentify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Long solution 💡 3 insights 📊 Diagram

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Problem

A corner of a tiled floor is shown. The same pattern continues across the whole floor, and each of the four corners of the floor looks like the one in the picture. Find the fraction of the floor that is covered by the darker tiles.

Pick an answer.

(A)

$\frac{1}3$

(B)

$\frac{4}9$

(C)

$\frac{1}2$

(D)

$\frac{5}9$

(E)

$\frac{5}8$

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The floor is infinite in the imagination but its tiling is periodic, so Tool #5 (Look for a Pattern) tells us the dark fraction is determined by one repeating block — we never have to count the whole floor. Tool #9 (Solve an Easier Related Problem) then shrinks the work even further: instead of working with the full corner shown, find the smallest square block that already captures the proportion. A small 3 × 3 block at the corner does the job. Tool #1 (Draw a Diagram) — really, reading the diagram already given — is what lets us mark each cell of that 3 × 3 block as dark or light and count.

1STEP 1

Because the same pattern repeats across the floor, the dark fraction of one repeating block equals the dark fraction of the whole floor.

(dark on floor)/(total on floor) = (dark in one block)/(total in one block)

2STEP 2

Look at the 3 × 3 square of tiles in the very corner; since all four corners match, its dark fraction equals the whole floor's.

block size = 3 × 3 = 9 tiles

3STEP 3

Read each cell of the 3 × 3 corner block from the diagram and mark it dark (D) or light (L), row by row from the top.

L & L & D ; L & D & D ; D & L & L ;

4STEP 4

Row 1 has 1 dark, row 2 has 2, row 3 has 1, so the block holds 4 dark tiles.

dark count = 1 + 2 + 1 = 4

5STEP 5

Divide dark tiles by the block's total tiles: 4 out of 9 is the whole floor's dark fraction — choice (B).

dark/total =

\frac{4}{9}

→ (B)

Answer

\frac{4}{9}

Check that the choice

\frac{4}{9}

fits the picture. A little less than half of the corner block is dark, which matches what the eye sees: the dark pinwheel covers most of one row, just a corner of another, and almost none of a third. The next-larger choice

\frac{1}{2}

would mean exactly half dark — clearly too much; the next-smaller choice

\frac{1}{3}

would mean only 3 dark out of 9 — clearly too few. The four-corner symmetry condition also rules out anything other than a single repeating block, so the small-block count is enough.

💡Key takeaway

When a pattern repeats, you don't need to count the whole floor — count one small block. A 3 × 3 corner has 4 dark tiles out of 9, so the answer is $\frac{4}{9}$ .

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