Problem 18 · 2021 Math Kangaroo
Stretch
Number Theory
sum-constraintcasework
7 cards are arranged as shown. Each card has 2 numbers on it, with 1 of them written upside down. The teacher wants to rearrange the cards so that the sum of the numbers in the top row is the same as the sum of the numbers in the bottom row. She can do this by turning one of the cards upside down. Which card must she turn?
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Answer: E — G
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Hint 1 of 2
Add up the top row and add up the bottom row, then see how far apart the two totals are.
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Hint 2 of 2
Turning one card swaps its top and bottom numbers, so look for the card whose swap moves exactly half the gap from one row to the other.
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Approach: compare the two row totals and flip the one card that evens them out
- The top numbers add to 7+5+4+2+8+3+2 = 31 and the bottom numbers add to 4+3+5+5+7+7+4 = 35, a gap of 4 (the bottom is 4 bigger).
- All 14 numbers add to 66, so to make the rows equal each must be 66 / 2 = 33; the top needs to gain 2 and the bottom to lose 2.
- Flipping a card moves its top number down and its bottom number up, so we need a card whose bottom is 2 more than its top — that is card G (top 2, bottom 4).
- So she must turn card G (E).
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