Problem 18 · 2024 Math Kangaroo
Stretch
Algebra & Patterns
sum-constraintsubstitution
Annie wants to write the numbers 1 to 10 in the ten circles (see picture). Each circle should have a different number. She wants the sum of the four numbers along each line to be exactly 23. Which number does she have to write in the circle with the question mark?
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Answer: D — 7
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Hint 1 of 3
Add every number once: 1 + 2 + ... + 10 = 55, then think about what you get if you instead add up all the line-totals.
Still stuck? Show hint 2 →
Hint 2 of 3
Each line is 23, but the circles where lines cross get counted more than once, so the line-totals add up to more than 55 — the extra amount is exactly the crossing circles.
Still stuck? Show hint 3 →
Hint 3 of 3
Work out how much 'extra' the crossings contribute, then see what the question-mark circle must be to make its lines reach 23.
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Approach: total all numbers, then use the leftover at the crossing circles
- The ten numbers 1 through 10 add to 55.
- Adding the line-totals counts every plain circle once but each crossing circle more than once, so the line-totals add up to more than 55; that surplus tells you the sum sitting at the shared crossing circles.
- Filling in the circles so each line reaches 23 forces the question-mark circle to balance its line.
- The required number is 7.
Quick check
Whatever line passes through the question mark must total 23; once the other three circles on that line are placed from 1–10, the question mark is what is left to reach 23, and that value is 7.
Mark:
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