Problem 24 · 2015 Math Kangaroo
Stretch
Spatial & Visual Reasoning
spatial-reasoningsum-constraint
Maria writes a number on each face of the cube. Then, for each corner point of the cube, she adds the numbers on the faces which meet at that corner. (For corner B she adds the numbers on faces BCDA, BAEF and BFGC.) In this way she gets a total of 14 for corner C, 16 for corner D, and 24 for corner E. Which total does she get for corner F?
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Answer: C — 22
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Hint 1 of 2
Two corners at opposite ends of a space diagonal use all six faces between them, so their corner-sums add up to the same total every time.
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Hint 2 of 2
Pair the given corner with the unknown one along a space diagonal, and pair the other two the same way.
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Approach: use that opposite corners of the cube share the same total of all six faces
- A corner's number is the sum of its three meeting faces. Two corners on opposite ends of a space diagonal together touch all six faces exactly once, so each such pair has the same sum S (the total of all six faces).
- Corners C and E are opposite, and corners D and F are opposite, so C + E = D + F.
- Thus 14 + 24 = 16 + F, giving F = 38 − 16 = 22.
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