Problem 11 · 2022 Math Kangaroo
Medium
Spatial & Visual Reasoning
cube-viewscareful-counting
Marc builds the number 2022 from 66 cubes of the same size, all glued together (see picture). He then paints the entire outer surface. On how many of the 66 cubes has Marc painted exactly four faces?
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Answer: E — 60
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Hint 1 of 2
Every face glued to a neighbouring cube is hidden; all the rest get painted.
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Hint 2 of 2
Since the digits are one cube thick, a cube shows 4 painted faces exactly when it touches 2 neighbours — so look for the few cubes that touch only 1.
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Approach: count cubes that touch exactly two neighbours
- The digits are one cube thick, so every cube always shows its front and back; it shows exactly 4 painted faces when it also has exactly 2 of its in-plane neighbours, i.e. it sits in a straight run or at a corner.
- The 0 is a closed loop, so every one of its cubes has 2 neighbours and shows 4 faces. Each 2 is an open strip with exactly two free ends, and those end cubes have only 1 neighbour, so they show 5 painted faces.
- The only exceptions are the 2 free ends on each of the three 2s, which is 6 cubes in all.
- That leaves 66 − 6 = 60 cubes with exactly four painted faces, so the answer is E.
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