AMC 10 · 2015 · #3

Grade 4 arithmetic

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

pattern-recognitionsequences-arithmetic easier-related-problem ↑ Prerequisites: pattern-recognition

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

A 3-step toothpick staircase is built from 18 toothpicks. Find how many more toothpicks must be added to turn it into a 5-step staircase.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Counting the whole 5-step figure from scratch is error-prone. Instead, track how the toothpick total grows by one step at a time. The jump from one staircase to the next follows a steady pattern, so I can extend that pattern from 3 steps up to 5 steps and add only the new pieces.

1STEP 1

Count smaller staircases

Count small staircases: 1-step=4, 2-step=10, 3-step (given)=18 toothpicks.

1-step=4, 2-step=10, 3-step=18

2STEP 2

Find the growth pattern

Each jump grows by 2: 1→2 step adds 6, 2→3 step adds 8 toothpicks.

+6, +8, … (each jump is 2 more than the last)

3STEP 3

Extend to steps 4 and 5

Continue the pattern: 3→4 step adds 10, 4→5 step adds 12 toothpicks.

3→4: +10, 4→5: +12

4STEP 4

Add the two jumps

Add both jumps: 10 + 12 = 22 toothpicks — choice (D).

10 + 12 = 22

Answer

Check against a direct count: an n-step staircase uses (n+1)(n+2) - 2 toothpicks, giving 18 for n=3 and (6)(7) - 2 = 40 for n=5. Then 40 - 18 = 22, matching the answer. 22 also sits sensibly between the choices 20 and 24.

💡Key takeaway

When a shape grows step by step, watch how much each step adds, then add up only the new pieces.

Count smaller staircases
Find the growth pattern
Extend to steps 4 and 5
Add the two jumps

Count smaller staircases

Find the growth pattern

Extend to steps 4 and 5

Add the two jumps

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