Competition · AMC preparation · step 4 of 4

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

Try a smaller grid first

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

Turn the pattern into a rule

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

Breaking the count into 'horizontal pile' and 'vertical pile' is the Tool #7 move — solve each piece, then add.

For a grid that is L cells long and W cells wide, the total number of toothpicks is (W+1) · L + (L+1) · W.

▸ Why?

Every toothpick lies either flat (horizontal) or upright (vertical) and none is both, so the total is just the horizontal count plus the vertical count.

▸ Why?

The horizontal pile and the vertical pile do not overlap and together cover every toothpick, so their two counts add back to the whole grid.

Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.

▸ Why?

The upright toothpicks form L+1 evenly spaced columns, and each column is a stack of W toothpicks, giving (L+1) · W.

▸ Why?

A straight run of L cells is walled off by L+1 vertical borders: one border on the left of each cell, plus one extra border to close the right end.

▸ Why?

Pair each cell with the border on its left — that matches the cells one-for-one with all but the last border, and the lone right-end border is the single leftover, so the borders number exactly one more than the cells.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

The L+1 columns are identical stacks of W toothpicks, so the count is L+1 equal groups of W, which is what multiplication measures.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

▸ Why?

The flat toothpicks form W+1 evenly spaced rows, and each row is a line of L toothpicks, giving (W+1) · L — the same fencepost and equal-groups reasoning with length and width swapped.

▸ Why?

A column of W stacked cells is walled off by W+1 horizontal borders: one below each cell, plus one extra to close the top end.

▸ Why?

Pair each cell with the border just below it — the cells match one-for-one with all but the top border, and that leftover top border makes the borders number exactly one more than the cells.

One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.

▸ Why?

The W+1 rows are identical lines of L toothpicks, so the count is W+1 equal groups of L, which is multiplication.

Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.

4.OA.A.3Identify Subproblems

3STEP 3

Count the vertical toothpicks

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

vertical = 61 × 32 = 1952

4STEP 4

Count the horizontal toothpicks

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

horizontal = 33 × 60 = 1980

5STEP 5

Add the two counts

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.

Try a smaller grid first
Turn the pattern into a rule
Count the vertical toothpicks
Count the horizontal toothpicks
Add the two counts

A parent dashboard for the family lives at sensimlab.com.

Try a smaller grid first

Turn the pattern into a rule

Count the vertical toothpicks

Count the horizontal toothpicks

Add the two counts

More like this