Problem 17 · 2012 Math Kangaroo
Stretch
Logic & Word Problems
sum-constraintcasework
The numbers 1 to 7 should be written in the small circles so that the sum of the numbers along each line is the same. Which number should be written in the uppermost circle on the triangle?
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Answer: C — 4
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Hint 1 of 3
First add up all the numbers you must place: 1 + 2 + ... + 7.
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Hint 2 of 3
The three corner circles each sit on two lines, so they get counted twice when you add the three line-sums together.
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Hint 3 of 3
Try giving the lines the smallest equal sum that works, and see which number is forced into the top circle.
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Approach: balance the three equal line-sums
- The seven numbers 1 to 7 add up to 28.
- Add the three line-totals together: every circle is counted, but the three corner circles each lie on two lines, so they get counted one extra time; the grand total is 28 plus the three corner numbers.
- For the three lines to share one equal sum, that grand total must split evenly by 3, and trying the natural balanced arrangement makes each line add to 10.
- Filling that in, the only number that lands in the top circle is 4 (choice C).
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