AMC 8 · 2004 · #15

Grade 4 counting

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence

pattern-recognitionsystematic-enumerationspatial-visualization pattern-recognitionsystematic-enumeration ↑ Prerequisites: multi-digit-arithmeticspatial-visualization

📏 Short solution 💡 2 insights 📊 Diagram

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Problem

A figure is built from 13 black and 6 white hexagonal tiles. A border of new white hexagons (same size and shape) is added around the outside. After the border is added, what is (total white tiles) - (total black tiles)?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The figure is built in concentric hexagonal rings: a center tile, then a ring of 6, then a ring of 12. The new border is just the next ring out. Tool #5 (Look for a Pattern) is the right primary — once we see the ring counts 6, 12, … jumping by 6, the next ring is forced to be 18. Tool #2 (Make a Systematic List) lays the ring counts and colors in a small table so the final tally — new whites added, blacks unchanged — is one clean subtraction.

1STEP 1

Split the figure into a center plus rings: 1 black center, 6 white in ring 1, 12 black in ring 2 — matching the given 13 and 6.

ring & tiles & color ; center & 1 & black ; ring 1 & 6 & white ; ring 2 & 12 & black black = 1+12 = 13, white = 6 (matches the problem)

2STEP 2

Ring sizes go 6, 12, … climbing by 6 each time, so the new border is the third ring with 18 tiles.

6, 12, 18, … (ring n has 6n tiles) → ring 3 = 18

3STEP 3

Blacks stay at 13; the 6 old whites plus 18 new border tiles make 24 white.

black = 13 (unchanged), white = 6 + 18 = 24

4STEP 4

Subtract to finish: total white minus total black is the difference the question asks for.

24 - 13 = 11 → (C)

Answer

Sanity check the count by a different route. Total tiles after the border: 1 + 6 + 12 + 18 = 37. Of those, 13 are black, so 37 - 13 = 24 are white, and 24 - 13 = 11 — matches (C). Quick elimination of distractors: (A) 5 and (B) 7 are too small (we added 18 whites and no blacks, so the gap must grow by 18 from the original 6 - 13 = -7, landing at +11). (E) 18 is the ring size itself, not the difference. (D) 12 would need only 6 extra whites, but the new ring has 18. So (C) is the only consistent value.

💡Key takeaway

Each new hexagon ring grows by a fixed 6 tiles — once you spot the rule, the new border's size is forced, and the rest is one subtraction.

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