Problem 22 · 2010 Math Kangaroo
Hard
Logic & Word Problems
sum-constraint
In the figure there are nine regions inside the circles. The numbers 1 to 9 should be written in the regions so that the sum of the numbers in each circle is exactly 11. Which number has to go in the region with the question mark?
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Answer: B — 6
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Hint 1 of 2
Add up the per-circle sums and compare with 1+2+...+9.
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Hint 2 of 2
Regions shared by two circles get counted twice — that overcount tells you the overlaps.
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Approach: double-count the circle sums
- The five circle-sums total 5×11 = 55, while the numbers 1–9 total 45.
- The extra 10 is the sum of the four shared (overlap) regions, which fixes the numbers around them.
- Filling the regions consistently puts 6 in the marked spot.
Mark:
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