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AMC 8 · 2013 · #22

Grade 4 countingpattern
Archetype Lattice / Grid Pattern from Small Cases
pattern-recognitionsystematic-enumerationmulti-digit-arithmetic pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticpattern-recognition
📏 Medium solution 💡 3 insights 📊 Diagram
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Problem

Toothpicks are used to make a grid that is 6060 toothpicks long and 3232 toothpicks wide. How many toothpicks are used altogether?

Pick an answer.

(A)
1920
(B)
1952
(C)
1980
(D)
2013
(E)
3932
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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Grid length $= 60$ toothpicks (number of cells in the horizontal direction); Grid width $= 32$ toothpicks (number of cells in the vertical direction); Each side of every unit cell is exactly one toothpick (shared toothpicks are NOT double-counted); Answer choices: (A) $1920$, (B) $1952$, (C) $1980$, (D) $2013$, (E) $3932$

Unknowns: The total number of toothpicks used to build the entire grid

Understand

Restated: 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?

Givens: Grid length $= 60$ toothpicks (number of cells in the horizontal direction); Grid width $= 32$ toothpicks (number of cells in the vertical direction); Each side of every unit cell is exactly one toothpick (shared toothpicks are NOT double-counted); Answer choices: (A) $1920$, (B) $1952$, (C) $1980$, (D) $2013$, (E) $3932$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #9 Solve an Easier Related Problem, #1 Draw a Diagram

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 $2 \times 3$ 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).

Execute — Answer: E

#9 Solve an Easier Related Problem 4.OA.C.5 Step 1
  • Start with an easier related problem: a grid that is $3$ toothpicks long and $2$ toothpicks wide.
  • Draw it and count by hand.
  • There are $3$ rows of horizontal toothpicks $+ 1$ extra row on top $= 3$ horizontal lines, each with $3$ toothpicks, giving $3 \times 3 = 9$ horizontal toothpicks.
  • There are $2$ columns $+ 1$ extra on the right $= 4$ vertical lines, each with $2$ toothpicks, giving $4 \times 2 = 8$ vertical toothpicks.
  • Total $17$.
$$\text{horizontal} = (2+1) \times 3 = 9, \quad \text{vertical} = (3+1) \times 2 = 8, \quad \text{total} = 17$$

💡 Building a tiny version makes the pattern '$n$ cells need $n+1$ lines' obvious — it's the fencepost rule, a Grade 4 pattern-recognition skill.

#7 Identify Subproblems 4.OA.A.3 Step 2
  • Generalize the rule from the small case.
  • For a grid that is $L$ toothpicks long and $W$ toothpicks wide: horizontal toothpicks $= (W+1) \times L$ and vertical toothpicks $= (L+1) \times W$.
  • This is the subproblem split — one count for each direction.
$$\text{Total} = (W+1) \cdot L + (L+1) \cdot W$$

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

#7 Identify Subproblems 4.NBT.B.5 Step 3
  • Count the vertical toothpicks for the real grid ($L = 60$, $W = 32$).
  • There are $60 + 1 = 61$ vertical lines (one extra for the right edge), and each line is $32$ toothpicks tall.
$$\text{vertical} = 61 \times 32 = 1952$$

💡 Multiplying a 2-digit number by a 2-digit number is the Grade 4 multi-digit multiplication standard.

#7 Identify Subproblems 4.NBT.B.5 Step 4
  • Count the horizontal toothpicks.
  • There are $32 + 1 = 33$ horizontal lines (one extra for the top edge), and each line is $60$ toothpicks long.
$$\text{horizontal} = 33 \times 60 = 1980$$

💡 Same kind of multi-digit multiplication as the vertical count, just with the roles of length and width swapped.

#7 Identify Subproblems 3.OA.A.1 Step 5
  • Add the two piles to get the total number of toothpicks.
  • This is the final 'combine' step of the subproblem decomposition.
$$1952 + 1980 = 3932 \;\Rightarrow\; \textbf{(E)}$$

💡 Adding the two products is the last step of the Tool #7 plan — solve each piece, then combine them.

[1] #9 4.OA.C.5 Start with an easier related problem: a grid that is $3$ toothpicks long and $2$
[2] #7 4.OA.A.3 Generalize the rule from the small case. For a grid that is $L$ toothpicks long
[3] #7 4.NBT.B.5 Count the vertical toothpicks for the real grid ($L = 60$, $W = 32$). There are
[4] #7 4.NBT.B.5 Count the horizontal toothpicks. There are $32 + 1 = 33$ horizontal lines (one e
[5] #7 3.OA.A.1 Add the two piles to get the total number of toothpicks. This is the final 'comb

Review

Reasonableness: Quick sanity check: a $60 \times 32$ grid has $60 \times 32 = 1920$ cells, and each cell would use $4$ toothpicks if nothing were shared, giving an upper bound of $1920 \times 4 = 7680$. 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.

Alternative: Tool #1 (Draw a Diagram) + Tool #5 (Look for a Pattern): sketch the smallest grids ($1\times1$, $2\times1$, $2\times2$, $3\times2$) and tabulate the totals. The pattern $T(L, W) = (W+1)L + (L+1)W = 2LW + L + W$ emerges, so $T(60, 32) = 2(60)(32) + 60 + 32 = 3840 + 92 = 3932$, matching choice (E).

CCSS standards used (min grade 4)

  • 3.OA.A.1 Interpret products of whole numbers (Reading $61 \times 32$ and $33 \times 60$ as 'how many toothpicks in each group of lines, times the number of lines,' and adding the two products at the end.)
  • 4.OA.A.3 Solve multistep word problems with the four operations (Splitting the grid into a 'horizontal pile' and a 'vertical pile,' counting each, and combining — a classic multistep word-problem structure.)
  • 4.OA.C.5 Generate and analyze patterns (Recognizing from the easier $3 \times 2$ grid that $n$ cells need $n+1$ lines (the fencepost pattern) and extending this rule to $L = 60$ and $W = 32$.)
  • 4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit number, and multiply two two-digit numbers (Computing the two-by-two-digit products $61 \times 32 = 1952$ and $33 \times 60 = 1980$.)

⭐ 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.

⭐ 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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