Problem 13 · 2012 Math Kangaroo
Medium
Spatial & Visual Reasoning
cube-viewsspatial-reasoning
Lisa built a large cube out of 8 smaller ones. The small cubes have the same letter on each of their faces (A, B, C or D). Two cubes with a common face always have a different letter on them. Which letter is on the cube that cannot be seen in the picture?
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Answer: B — B
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Hint 1 of 3
There are 8 little cubes but only 4 letters, and every cube touches three neighbours that must all differ from it.
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Hint 2 of 3
Two cubes can share a letter only if they do NOT touch, i.e. they sit at opposite ends of a long diagonal through the centre.
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Hint 3 of 3
The hidden cube is the corner diagonally opposite a visible one, so it copies that cube's letter.
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Approach: opposite corners share a letter
- Each small cube touches 3 others (one in each direction), and touching cubes must differ, so a cube and its 3 neighbours use up all 4 letters A, B, C, D.
- That means a letter can repeat only on two cubes that never touch, namely the two ends of a diagonal running through the centre of the big cube.
- So each of the 4 space-diagonals carries one repeated letter, pairing every cube with the corner diagonally across from it.
- The unseen back corner is diagonally opposite a visible corner, and matching their letter gives the hidden one: B.
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