Problem 16 · 2014 Math Kangaroo
Medium
Spatial & Visual Reasoning
cube-viewssum-constraint
The vertices of a die are numbered 1 to 8 so that the sum of the four numbers on the vertices of each face is the same. The numbers 1, 4 and 6 are already indicated in the picture. Which number is in position x?
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Answer: A — 2
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Hint 1 of 2
Every face uses 4 of the 8 corner numbers and they must all add to the same total.
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Hint 2 of 2
Total of 1…8 is 36; use that to find each face's required sum, then fill in around the given 1, 4, 6.
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Approach: fix the common face sum, then deduce the corners
- The eight corner labels add to 1+…+8 = 36; pairing opposite faces shows each face must sum to 18.
- Placing the given 1, 4 and 6 and forcing every face to total 18 determines all remaining corners uniquely.
- The corner at position x comes out to 2.
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