Problem 11 · 1999 AMC 8
Medium
Logic & Word Problems
counted-twiceoptimization
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Answer: D — 24.
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Hint 1 of 2
The plus shape overlaps at the center square — that one number belongs to BOTH the row and the column. So if you add the row-sum and the column-sum together, the center gets counted twice while everyone else is counted once.
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Hint 2 of 2
Equal sums means each line is exactly half of (row + column). To make that half as big as possible, you want the doubled number — the center — to be the largest. Name this the 'shared-cell counts twice' idea.
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Approach: the overlap cell is counted twice — load it to maximize
- Add the row total and the column total. Every number appears once except the center, which sits in both lines, so the grand total is (1+4+7+10+13) + center = 35 + center.
- The two lines are equal, so each equals half of 35 + center. To push that as high as possible, put the biggest number, 13, in the center: (35 + 13) ÷ 2 = 48 ÷ 2 = 24.
- Worth keeping: whenever a cell is shared by two groups whose sums you're combining, it gets double-counted — and that's a lever. Want the largest equal sums? Maximize the shared cell. (Sanity check: 24 must be achievable — center 13, then pair the rest as 1+10 and 4+7, each making 11, and 11+13 = 24. ✓)
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