AMC 10 · 2008 · #2
Grade 2 arithmeticPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The whole difficulty is keeping track of which number sits in which cell after two rows flip, so Tool #1 (Draw a Diagram) — literally rewriting the block with rows 2 and 4 reversed — removes every chance of grabbing the wrong number off a diagonal. Once the new block is drawn, Tool #7 (Identify Subproblems) splits the job into three tiny, independent calculations: sum one diagonal, sum the other, subtract.
Redraw the block after reversing
Rows 1 and 3 stay put; row 2 flips to 11,10,9,8 and row 4 flips to 25,24,23,22.
Draw the board exactly as it looks after the flip so you read real numbers off it, not remembered ones.
Draw the board exactly as it looks after the flip, so you read real numbers off it rather than remembered ones.
▸ Why?
Flipping a row moves the numbers without changing any of them, so the new board is fully determined.
▸ Why?
Each old position lands on exactly one new position, so nothing is lost or duplicated in the redraw.
Add the main diagonal
Going top-left to bottom-right you pass 1, 10, 17, 22, and their sum is 50.
Step down the diagonal one row at a time, moving one column right each row.
2.NBT.B.6Identify SubproblemsAdd the other diagonal
Going top-right to bottom-left you pass 4, 9, 16, 25, and their sum is 54.
This diagonal starts at the top-right corner and steps down-left, one column left each row.
2.NBT.B.6Identify SubproblemsTake the positive difference
The larger total minus the smaller gives 54-50=4, which is choice (B).
Positive difference just means subtract the smaller sum from the larger one.
2.NBT.B.5Identify SubproblemsWhen numbers move around a grid, redraw the grid first and then read the diagonals off the new picture instead of your memory.
- Redraw the block after reversing
- Add the main diagonal
- Add the other diagonal
- Take the positive difference
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