AMC 10 · 2004 · #8
Grade 4 arithmeticPick an answer.
Because each row grows by the same step of 2, the row sizes are the odd numbers 1, 3, 5, 7, …, 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, …, which are perfect squares, so the total after n rows is n × 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.
Read the rows as odd numbers
The rows read 1, 3, 5, 7, … — the odd numbers in order.
Stepping up by 2 every time walks you straight through the odd numbers.
4.OA.C.5Look For A PatternAdd row by row and watch the totals
The running totals are 1, 4, 9, 16, … — always a perfect square.
Each new odd row is an L-shape that hugs the current square and turns it into the next bigger square.
Each new row is an L-shape wrapped around the current square, turning it into the next bigger square.
▸ Why?
The rows step up by the same two seats each time, so they run straight through the odd numbers.
▸ Why?
Pairing the first row with the last gives the same total as pairing inward, so the running total is always a square.
Match the total to 100
So the total 100 needs the square root, giving 10 rows, choice (D).
Reaching a total of 100 means reaching the square 10 × 10, which happens at exactly 10 rows.
3.OA.C.7Guess And CheckAdding up the odd numbers 1+3+5+… always gives a perfect square, so a stack that grows by 2 cans per row reaches 100=10 × 10 after exactly 10 rows.
- Read the rows as odd numbers
- Add row by row and watch the totals
- Match the total to 100