Problem 19 · 2016 Math Kangaroo
Medium
Spatial & Visual Reasoning
dice-faces
Seven identical dice (each with 1, 2, 3, 4, 5 and 6 points on its faces) are glued together to form the solid shown. Faces that are glued together always show the same number of points. How many points can be seen on the surface of the solid?
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Answer: D — 105
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Hint 1 of 3
Each die has 1 + 2 + ... + 6 = 21 points; seven dice hold 7 × 21.
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Hint 2 of 3
Subtract the hidden glued faces, which come in equal-number pairs.
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Hint 3 of 3
The six gluings hide pairs that total a fixed amount; remove it.
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Approach: total points minus the hidden glued faces
- Seven dice have 7 × 21 = 147 points in all.
- The central die is glued to 6 others; each gluing hides two equal faces, and over the 6 contacts the hidden faces sum to 42.
- Visible points = 147 − 42 = 105.
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