Problem 14 · 2023 Math Kangaroo
Medium
Arithmetic & Operations
sum-constraint
The sum of the numbers in the white fields should equal the sum of the numbers in the grey fields. Which two numbers have to be swapped so that the two sums become equal?
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Answer: A — 1 and 11
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Hint 1 of 2
Add up the grey numbers and the white numbers separately to see the gap between them.
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Hint 2 of 2
A swap moves one number out of each group; close exactly half the gap by picking the right pair.
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Approach: balance the two totals by closing half the gap
- The grey fields total 1 + 2 + 7 + 4 + 6 = 20 and the white fields total 3 + 5 + 13 + 8 + 11 = 40, so the whole grid sums to 60 and each side should reach 30.
- Swapping a grey number g for a white number w changes the grey total by (w − g); to go from 20 to 30 we need w − g = 10.
- Swapping grey 1 with white 11 gives w − g = 10, making both sides 30.
- So swap 1 and 11, answer A.
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