AMC 10 · 2018 · #8
Easy mode Grade 4Sara builds a staircase shape out of toothpicks, like the one in the picture.
The staircase in the picture has 3 steps and uses 18 toothpicks. A bigger staircase, built the same way, uses 180 toothpicks. How many steps does it have?
Pick an answer.
AMC 10 2018 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: A staircase is built from toothpicks. A $3$-step staircase uses $18$ toothpicks. Find how many steps a staircase has when it uses $180$ toothpicks.
Givens: A $3$-step staircase uses exactly $18$ toothpicks; Each step adds another row of unit squares made of toothpicks; A larger staircase uses $180$ toothpicks; Answer choices: (A) $10$, (B) $11$, (C) $12$, (D) $24$, (E) $30$
Unknowns: The number of steps in the staircase that uses $180$ toothpicks
Understand
Restated: A staircase is built from toothpicks. A $3$-step staircase uses $18$ toothpicks. Find how many steps a staircase has when it uses $180$ toothpicks.
Givens: A $3$-step staircase uses exactly $18$ toothpicks; Each step adds another row of unit squares made of toothpicks; A larger staircase uses $180$ toothpicks; Answer choices: (A) $10$, (B) $11$, (C) $12$, (D) $24$, (E) $30$
Plan
Primary tool: #9 Solve an Easier Related Problem
Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities
Jumping straight to $180$ toothpicks is hard, so Tool #9 (Solve an Easier Related Problem) leads: count the toothpicks in tiny staircases of $1$, $2$, and $3$ steps where counting is easy. Tool #5 (Look for a Pattern) then reads the rule out of those small counts — the totals grow by a step that itself grows by $2$ each time. Tool #3 (Eliminate Possibilities) finishes the multiple-choice problem: extend the pattern and see which step number lands on exactly $180$.
Execute — Answer: C
4.OA.C.5 Step 1 Count the smallest staircases
- Start tiny.
- A $1$-step staircase is one square: $4$ toothpicks.
- A $2$-step staircase is two squares on the bottom with one square stacked on the left; counting the shared edges only once gives $10$ toothpicks.
- The given $3$-step staircase is $18$ toothpicks.
- So the counts begin $4,\ 10,\ 18$.
💡 Big figures are made of the same kind of small figure, so the rule shows up clearly in sizes you can count by hand.
3.OA.D.9 Step 2 Read the growth rule
- Look at how much is added each time.
- From $4$ to $10$ is $+6$; from $10$ to $18$ is $+8$.
- The jump itself grows by $2$ every step, so the next jumps are $+10$, $+12$, $+14$, and so on.
💡 Each new step glues on one more row of squares than the last, so the number of fresh toothpicks climbs by a steady $2$ each time.
4.NBT.B.4 Step 3 Extend the pattern toward 180
- Keep adding the growing jumps until the total reaches $180$: $4,\ 10,\ 18,\ 28,\ 40,\ 54,\ 70,\ 88,\ 108,\ 130,\ 154,\ 180$.
- Counting positions, $180$ is the $12$th number in the list.
💡 If you trust the rule, you can just keep stacking on the next jump and watch for the target total to appear.
4.NBT.B.4 Step 4 Check it against the choices
- The $10$-step staircase needs $130$ toothpicks and the $11$-step needs $154$ — both short of $180$ — while $12$ steps land on exactly $180$.
- So choices (A) and (B) are too small, and the larger choices $24$ and $30$ would overshoot $180$ badly.
- The number of steps is $12$, which is choice (C).
💡 Only one step-count can hit $180$ exactly, so matching the total to the list pins down the answer.
4.OA.C.5 Start tiny. A $1$-step staircase is one square: $4$ toothpicks. A $2$-step stair 3.OA.D.9 Look at how much is added each time. From $4$ to $10$ is $+6$; from $10$ to $18$ 4.NBT.B.4 Keep adding the growing jumps until the total reaches $180$: $4,\ 10,\ 18,\ 28,\ 4.NBT.B.4 The $10$-step staircase needs $130$ toothpicks and the $11$-step needs $154$ — b Review
Reasonableness: The total $180$ is exactly $10$ times the $3$-step total of $18$, but the answer is $12$ steps, not $30$ — that fits, because the toothpicks grow faster than the step number, so far fewer steps are needed than a simple $\times 10$ would suggest. This rules out the trap choices (D) $24$ and (E) $30$. The found total $T(12)=180$ matches the target precisely.
Alternative: Tool #13 (Convert to Algebra): the small counts fit $T(n)=n(n+3)$, since $1\cdot4=4$, $2\cdot5=10$, $3\cdot6=18$. Setting $n(n+3)=180$ gives $n^2+3n-180=0$, which factors as $(n-12)(n+15)=0$. The positive root is $n=12$ — the same answer (C).
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern following a given rule (Counting the $1$-, $2$-, and $3$-step staircases to get the starting numbers $4$, $10$, $18$.)3.OA.D.9Identify arithmetic patterns and explain using properties of operations (Spotting that the amount added each step ($+6,+8,+10,\dots$) itself grows by a constant $2$.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Adding the growing jumps to extend the list to $180$ and confirming $T(12)=180$ against the answer choices.)
⭐ When a figure is too big to count, count the tiny ones first, find how the totals grow, and walk the pattern up to the number you need.
⭐ When a figure is too big to count, count the tiny ones first, find how the totals grow, and walk the pattern up to the number you need.
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