Competition · AMC preparation · step 4 of 4

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

Reduce to one block

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)

Grade 4 "generate and analyze a repeating pattern" — once the unit repeats, every copy looks the same, so any one copy answers the whole-floor question.

The fraction of tiles that are dark inside one repeating block is the same as the fraction of the whole floor that is dark.

▸ Why?

The whole floor is nothing but many identical copies of that one block, so its dark tiles and its total tiles are each just the block's count taken as many times as there are copies.

▸ Why?

Because the pattern repeats, every part of the floor is the same block slid into a new spot, and sliding the block lays it exactly onto its copy, so the dark tiles land on dark tiles and each copy keeps the very same dark count and the very same total.

Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).

▸ Why?

The copies fill the floor with no gaps and no overlaps, so the floor's dark tiles are exactly all the copies' dark tiles gathered back together, and with equal copies that gathering is the block's dark count taken once per copy.

▸ Why?

Fitting the copies edge to edge with nothing left uncovered and nothing counted twice means the floor's tiles are precisely the copies' tiles put back together into the whole.

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?

Adding the block's count once for each copy, over and over the same amount, is the same as taking that many equal groups of it, which is that count multiplied by the number of copies.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

So the floor's dark-over-total is the block's dark times the number of copies over the block's total times that same number of copies — that is the block's own fraction with its top and its bottom each scaled up by the same number of copies, and scaling a fraction's top and bottom by the same number leaves its value unchanged, so the floor's fraction is exactly the block's.

Equivalent fractionsMultiplying (or dividing) a fraction's top and bottom by the same nonzero number leaves its value unchanged.

4.OA.C.5Look For A Pattern

2STEP 2

Pick the smallest block

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

Mark each cell dark or light

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

Count the dark cells

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

Form the fraction

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

dark/total = 4/9 → (B)

Answer

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 $\frac{4}{9}$ .

Reduce to one block
Pick the smallest block
Mark each cell dark or light
Count the dark cells
Form the fraction

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Reduce to one block

Pick the smallest block

Mark each cell dark or light

Count the dark cells

Form the fraction

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