AMC 10 · 2004 · #10
Grade 4 arithmeticPick an answer.
AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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
Start at the top with 1 can and add 2 going down: 1, 3, 5, 7, 9, … — 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
Running totals are 1, 4, 9, 16, 25 — perfect squares. So n rows hold n × n cans.
Each new odd row is an L-shape that hugs the current square and turns it into the next bigger square.
Each new odd row is an L-shape that hugs the current square and turns 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 a square.
Match the total to 100
The stack holds 100 cans, so n × n = 100. Since 10 × 10 = 100, the count is n = 10 rows — (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