Problem 20 · 2020 Math Kangaroo
Stretch
Number Theory
sum-constraintcasework
Six different numbers, chosen from the whole numbers 1 to 9, are written on the faces of a cube, one number per face. The sum of the two numbers on each pair of opposite faces is always the same. Which of these numbers could be written on the face opposite the number 8?
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Answer: A — 3
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Hint 1 of 2
Opposite faces share one common sum; three of the digits 1-9 are left off.
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Hint 2 of 2
Whatever sits opposite 8 must give that same pair-sum as the other two pairs.
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Approach: find a consistent equal pair-sum
- The picture shows 4, 5 and 8 on three faces that touch, so none of them is opposite another; their partners are the three hidden faces.
- All three pairs share one common sum. If that sum is 11, then 8 pairs with 3, 5 pairs with 6, and 4 pairs with 7 — the six numbers 3, 4, 5, 6, 7, 8 are all different and all between 1 and 9.
- Any larger common sum forces a partner above 9, so the only choice that works is 8 opposite 3.
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