AMC 8 · 2013 · #22
Grade 4 countingpatternProblem
Toothpicks are used to make a grid that is 60 toothpicks long and 32 toothpicks wide. How many toothpicks are used altogether?
Pick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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 $17$.
💡 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.
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.
💡 Breaking the count into 'horizontal pile' and 'vertical pile' is the Tool #7 move — solve each piece, then add.
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.
💡 Multiplying a 2-digit number by a 2-digit number is the Grade 4 multi-digit multiplication standard.
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.
💡 Same kind of multi-digit multiplication as the vertical count, just with the roles of length and width swapped.
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.
💡 Adding the two products is the last step of the Tool #7 plan — solve each piece, then combine them.
4.OA.C.5 Start with an easier related problem: a grid that is $3$ toothpicks long and $2$ 4.OA.A.3 Generalize the rule from the small case. For a grid that is $L$ toothpicks long 4.NBT.B.5 Count the vertical toothpicks for the real grid ($L = 60$, $W = 32$). There are 4.NBT.B.5 Count the horizontal toothpicks. There are $32 + 1 = 33$ horizontal lines (one e 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.1Interpret 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.3Solve 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.5Generate 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.5Multiply 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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