AMC 10 · 2008 · #7

Grade 6 geometry-2d
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
similar-figuresarea-trianglesperfect-squares easier-related-problem ↑ Prerequisites: area-triangles
📏 Medium solution 💡 2 insights
Problem
A big equilateral triangle has side length 10. It is packed completely, with no gaps and no overlaps, by little equilateral triangles that each have side length 1. Count how many little triangles that takes.

Pick an answer.

(A)
$\ 10$
(B)
$\ 25$
(C)
$\ 100$
(D)
$\ 250$
(E)
$\ 1000$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Side length 10 is hard to picture all at once, so shrink the problem first (Tool #9): try side 1, side 2, side 3 and actually draw each one (Tool #1). Counting those small cases gives 1, 4, 9 — the perfect squares. That pattern (Tool #5) says a side-n triangle needs n² pieces, and then side 10 is just 10². The pattern route is safer than juggling the area formula, because you can literally see and check each small case.

1STEP 1

Split a small triangle

Start small: a side-2 triangle cuts into 3 pieces pointing up and 1 pointing down — 4 unit triangles; a side-3 triangle gives 9.

side 2 → 4, side 3 → 9
2STEP 2

Count row by row

Count by rows from the bottom: a side-n triangle reads 2n-1, 2n-3, …, 3, 1, dropping by 2 each row. Side 3 gives 5+3+1=9.

side n: (2n-1)+(2n-3)+…+3+1
3STEP 3

Spot the square

The totals 1, 4, 9, 16 are exactly 1², 2², 3², 4², so a side-n triangle needs n² unit triangles.

1,4,9,16 = 1²,2²,3²,4² → side n → n²
4STEP 4

Apply it to side 10

Put n=10 into the rule: 10² = 10 × 10 = 100, choice (C). The count squares with the side, so the tempting 10 is out.

n=10: 10² = 100 → (C)
Answer
100
The answer 100 passes an independent area check. Similar triangles scale area by the square of the length ratio, and the big triangle is 10 times as long on each side as a unit triangle, so its area is 10²=100 times as large. Equal-area pieces that tile it perfectly must therefore number 100. Both the row-by-row count and the area argument land on the same value, and 100 is choice (C). The distractors are the classic traps: 10 (forgetting to square), 250 and 1000 (over-scaling).
💡Key takeaway

A triangle with side n made of unit triangles needs n × n of them, so side 10 takes 10² = 100.

  • Split a small triangle
  • Count row by row
  • Spot the square
  • Apply it to side 10

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