AMC 8 · 2013 · #22

Grade 4 countingpattern

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

pattern-recognitionsystematic-enumerationmulti-digit-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticpattern-recognition

📏 Medium solution 💡 3 insights 📊 Diagram

📘 View easy version →

Problem

Toothpicks are laid end-to-end to form a rectangular grid that is 60 toothpicks long (horizontal direction) and 32 toothpicks wide (vertical direction). Every small cell of the grid is bordered by single toothpicks. How many toothpicks are used in the whole grid?

Pick an answer.

(A)

1920

(B)

1952

(C)

1980

(D)

2013

(E)

3932

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Counting all 3932 toothpicks at once is hopeless, so Tool #7 (Identify Subproblems) splits the grid into two clean piles: every toothpick is either horizontal or vertical, and the two piles can be counted separately and added. To see why a grid that is n toothpicks long needs n+1 vertical lines, Tool #9 (Easier Related Problem) is perfect — shrink to a tiny 2 × 3 grid where you can count directly. Tool #1 (Draw a Diagram) just makes the "n+1 lines for n cells" fact visible (fenceposts vs. fence sections).

1STEP 1

Warm up on a tiny 3-long, 2-wide grid: draw it and count by hand to get 17 toothpicks.

horizontal = (2+1) × 3 = 9, vertical = (3+1) × 2 = 8, total = 17

2STEP 2

Generalize: split into two piles — horizontal = (W+1) × L, vertical = (L+1) × W — where n cells always take n+1 lines.

Total = (W+1) · L + (L+1) · W

3STEP 3

Real grid: vertical lines number 60 + 1 = 61, each 32 tall, so vertical toothpicks = 1952.

vertical = 61 × 32 = 1952

4STEP 4

Now horizontal: lines number 32 + 1 = 33, each 60 long, so horizontal toothpicks = 1980.

horizontal = 33 × 60 = 1980

5STEP 5

Add the two piles: 1952 + 1980 = 3932, choice (E).

1952 + 1980 = 3932 → (E)

Answer

3932

Quick sanity check: a 60 × 32 grid has 60 × 32 = 1920 cells, and each cell would use 4 toothpicks if nothing were shared, giving an upper bound of 1920 × 4 = 7680. But almost every interior edge is shared by two cells, so the true count should be a bit less than half of that — roughly 4000. Our answer 3932 lands right in that window. The trap answers 1920, 1952, 1980 are exactly the pieces we computed along the way (cell count, vertical count, horizontal count), which is why the problem includes them as distractors.

💡Key takeaway

This AMC 8 problem becomes simple once you split it in two — count horizontal toothpicks, count vertical toothpicks, add. The only new idea is the fencepost rule: n cells need n+1 lines, a Grade 4 pattern.

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