AMC 10 · 2018 · #8

Grade 4 arithmetic
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Figure-N Growth Sequence
pattern-recognitionsequences-arithmetic easier-related-problempattern-recognition ↑ Prerequisites: pattern-recognition
📏 Medium solution 💡 2 insights 📊 Diagram
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Problem
A staircase is built from toothpicks. A 3-step staircase uses 18 toothpicks. Find how many steps a staircase has when it uses 180 toothpicks.

Pick an answer.

(A)
10
(B)
11
(C)
12
(D)
24
(E)
30

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

Jumping straight to 180 toothpicks is hard, so Tool #9 (Solve an Easier Related Problem) leads: count the toothpicks in tiny staircases of 1, 2, and 3 steps where counting is easy. Tool #5 (Look for a Pattern) then reads the rule out of those small counts — the totals grow by a step that itself grows by 2 each time. Tool #3 (Eliminate Possibilities) finishes the multiple-choice problem: extend the pattern and see which step number lands on exactly 180.

1STEP 1

Count the smallest staircases

By hand: a 1-step staircase is 4 toothpicks, 2 steps is 10, and the given 3 steps is 18 — so the totals begin 4, 10, 18.

T(1)=4, T(2)=10, T(3)=18
2STEP 2

Read the growth rule

See how much is added: 4→10 is +6, 10→18 is +8. The jump itself grows by 2 each time, so next come +10, +12, +14.

4 → 10 → 18 → 28 → 40 …
3STEP 3

Extend the pattern toward 180

Keep adding the growing jumps: 4, 10, 18, 28, 40, 54, 70, 88, 108, 130, 154, 180 — and 180 is the 12th number.

4₁, 10₂, 18₃, 28, 40, 54, 70, 88, 108, 130₁0, 154₁1, 180₁2
4STEP 4

Check it against the choices

10 steps need 130 and 11 steps 154 — both short of 180 — while 12 steps hit exactly 180; 24 and 30 overshoot. The answer is (C).

T(10)=130, T(11)=154, T(12)=180 → (C)
Answer
12
The total 180 is exactly 10 times the 3-step total of 18, but the answer is 12 steps, not 30 — that fits, because the toothpicks grow faster than the step number, so far fewer steps are needed than a simple × 10 would suggest. This rules out the trap choices (D) 24 and (E) 30. The found total T(12)=180 matches the target precisely.
💡Key takeaway

When a figure is too big to count, count the tiny ones first, find how the totals grow, and walk the pattern up to the number you need.

  • Count the smallest staircases
  • Read the growth rule
  • Extend the pattern toward 180
  • Check it against the choices

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