AMC 10 · 2004 · #8
Easy mode Grade 4Cans are stacked in rows. The top row has 1 can. Each row below has 2 more cans than the row right above it. Altogether the stack uses 100 cans. How many rows does it have?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Cans are stacked in rows. The top row has $1$ can, and every row below holds $2$ more cans than the row directly above it. The whole stack uses $100$ cans. Find how many rows the stack has.
Givens: The top row has $1$ can.; Each row below has exactly $2$ more cans than the row above it.; The stack contains $100$ cans in total.; Answer choices: (A) $5$, (B) $8$, (C) $9$, (D) $10$, (E) $11$.
Unknowns: The number of rows in the stack.
Understand
Restated: Cans are stacked in rows. The top row has $1$ can, and every row below holds $2$ more cans than the row directly above it. The whole stack uses $100$ cans. Find how many rows the stack has.
Givens: The top row has $1$ can.; Each row below has exactly $2$ more cans than the row above it.; The stack contains $100$ cans in total.; Answer choices: (A) $5$, (B) $8$, (C) $9$, (D) $10$, (E) $11$.
Plan
Primary tool: #5 Look for a Pattern
Secondary: #1 Draw a Diagram, #6 Guess and Check
Because each row grows by the same step of $2$, the row sizes are the odd numbers $1, 3, 5, 7, \ldots$, and the running totals hide a clean pattern — the signature trigger for Tool #5 (Look for a Pattern). Adding the rows one at a time gives $1, 4, 9, 16, \ldots$, which are perfect squares, so the total after $n$ rows is $n\times n$. Tool #1 (Draw a Diagram) shows why: each new odd row wraps an L-shape around a square and completes the next larger square. Tool #6 (Guess and Check) then just asks which square equals $100$.
Execute — Answer: D
4.OA.C.5 Step 1 Read the rows as odd numbers
- Start at the top with $1$ can and add $2$ cans for each row going down.
- That gives $1$, then $1+2=3$, then $3+2=5$, then $5+2=7$, and so on.
- The number of cans in the rows, from top to bottom, is $1, 3, 5, 7, 9, \ldots$ — the odd numbers in order.
💡 Stepping up by $2$ every time walks you straight through the odd numbers.
3.OA.D.9 Step 2 Add row by row and watch the totals
- Add the rows one at a time and keep a running total.
- One row: $1$.
- Two rows: $1+3=4$.
- Three rows: $4+5=9$.
- Four rows: $9+7=16$.
- Five rows: $16+9=25$.
- The totals are $1, 4, 9, 16, 25$, which are exactly $1\times 1,\ 2\times 2,\ 3\times 3,\ 4\times 4,\ 5\times 5$.
- So the total after $n$ rows is $n\times n$.
💡 Each new odd row is an L-shape that hugs the current square and turns it into the next bigger square.
3.OA.C.7 Step 3 Match the total to 100
- The stack must total $100$ cans, and the total after $n$ rows is $n\times n$.
- So find the number that multiplied by itself gives $100$.
- Since $10\times 10=100$, the stack needs $n=10$ rows.
- The answer is (D).
💡 Reaching a total of $100$ means reaching the square $10\times 10$, which happens at exactly $10$ rows.
4.OA.C.5 Start at the top with $1$ can and add $2$ cans for each row going down. That giv 3.OA.D.9 Add the rows one at a time and keep a running total. One row: $1$. Two rows: $1+ 3.OA.C.7 The stack must total $100$ cans, and the total after $n$ rows is $n\times n$. So Review
Reasonableness: Check the neighbors: $9$ rows give $9\times 9=81$ cans and $11$ rows give $11\times 11=121$ cans, so only $10$ rows land exactly on $100$. The wrong choices match squares that miss: $5$ rows make $25$ and $8$ rows make $64$, both far short. Landing precisely on the offered choice $10$ is a strong sign the count is right.
Alternative: Use the arithmetic-series shortcut instead of spotting squares. With $n$ rows, the bottom row holds $2n-1$ cans, and the sum of the rows is $\tfrac{n}{2}\times(\text{first}+\text{last})=\tfrac{n}{2}\times(1+(2n-1))=\tfrac{n}{2}\times 2n=n\times n$. Setting $n\times n=100$ gives $n=10$ again. Or simply keep adding odd cans until the running total hits $100$: $1,4,9,16,25,36,49,64,81,100$ — that is $10$ additions, so $10$ rows.
CCSS standards used (min grade 4)
4.OA.C.5Generate a number or shape pattern that follows a given rule (Generating the row sizes $1, 3, 5, 7, \ldots$ from the rule 'start at 1 and add 2 each row.')3.OA.D.9Identify arithmetic patterns and explain them using properties of operations (Noticing that the running totals $1, 4, 9, 16, 25$ are the perfect squares, so the total after $n$ rows is $n\times n$.)3.OA.C.7Fluently multiply and divide within 100 (Recognizing that $10\times 10=100$ to find the number of rows.)
⭐ Adding up the odd numbers $1+3+5+\cdots$ always gives a perfect square, so a stack that grows by $2$ cans per row reaches $100=10\times 10$ after exactly $10$ rows.
⭐ Adding up the odd numbers $1+3+5+\cdots$ always gives a perfect square, so a stack that grows by $2$ cans per row reaches $100=10\times 10$ after exactly $10$ rows.
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