Problem 9 · 2020 AMC 8
Medium
Geometry & Measurement
spatial-reasoningcareful-counting
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Answer: D — 20 pieces.
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Hint 1 of 2
Two iced faces can only meet along an edge, so hunt along the cake's edges, not the flat faces. But beware: the bottom has no icing, which breaks the usual up-down symmetry — treat top edges and bottom edges differently.
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Hint 2 of 2
Sort the edges into kinds: 4 top edges (top + a side), 4 vertical corner edges (side + side), and 4 bottom edges (a side + the un-iced bottom = only 1 iced face, so they don't count). Subtract the top corners, which have 3 iced sides.
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Approach: classify cubes by exposed faces; respect the missing bottom icing
- A cube has two iced sides only where two iced faces meet — an edge. Top corners touch three iced faces (top + two sides), and the bottom face has no icing, so the bottom edges only ever touch one iced face. That leaves two qualifying edge types.
- Top edges (top + one side): each of the 4 has 4 cubes; the 2 ends are top corners (3 iced) → 2 good per edge → 4 × 2 = 8.
- Vertical corner edges (side + side): each of the 4 has 4 cubes; the top one is a top corner already counted, and the bottom 3 each touch 2 sides (bottom un-iced doesn't add a face) → 4 × 3 = 12.
- Total: 8 + 12 = 20 pieces.
- Why this transfers: “painted cube” problems are all about classifying small cubes by how many faces are exposed (corner = 3, edge = 2, face = 1). The trick here is that removing one face's icing destroys the top–bottom symmetry, so you can't just double the top — always recheck which faces actually count.
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