Problem 14 · 2022 Math Kangaroo
Hard
Spatial & Visual Reasoning
dice-faces
On an ordinary die the numbers on opposite faces always add up to 7. Four such dice are glued together as shown. All the numbers still visible on the outside of the solid are added up. What is the smallest possible value of that total?
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Answer: D — 58
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Hint 1 of 2
Opposite faces of a die sum to 7, so all six faces of one die total 21 and the four dice total 84.
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Hint 2 of 2
Visible total = 84 minus the glued (hidden) faces, so MINIMISE the visible sum by putting the biggest allowed numbers on the touching faces.
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Approach: subtract the hidden glued faces from the four-dice total
- The four dice have total pip count 4 x 21 = 84, and the visible sum is 84 minus whatever is hidden at the glued joints.
- To make the visible sum smallest, orient the dice so the touching faces carry as many pips as the gluing in the figure allows.
- Maximising the hidden pips this way leaves a smallest visible total of 58, so the answer is D.
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