Problem 1 · AMC 8 Stretch
Core
Geometry & Measurement
Logic & Word Problems
account-for-all-possibilitiesconsider-extreme-cases
A cube has six faces (flat surfaces). Now cut a cube in half. How many faces does half a cube have? Try to think of all the different ways the cube might be cut in half.
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Answer: No single answer: 6 for a flat straight cut, 5 for a slanted cut, more for a staircase cut — the point is to define what 'half' means
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Hint 1 of 4
There is no single 'right' answer here! The number of faces depends on HOW you slice. Start by picturing the simplest cut you can imagine.
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Hint 2 of 4
Try a straight cut straight down through the middle, like slicing a block of cheese. You get a smaller box. Count its flat surfaces carefully — don't forget the brand-new face the knife made.
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Hint 3 of 4
Now try a slanted cut, going corner to corner. Count again. Did you get a different number? What does that tell you?
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Approach: Open-ended exploration: the answer depends on the cut
- This problem is meant to be open-ended; its value is in the discussion, not in one magic number.
- Straight cut down the middle: you get a smaller rectangular box, which has 6 flat faces — so this answer is 6 (one is the new face the knife made).
- Slanted (corner-to-corner) cut: the piece is a wedge or triangular prism, and you count 5 faces. (Some people blurt out 3 before noticing the original faces are still partly there.)
- Staircase cut: if the cut is a jagged staircase that still splits the cube into two equal-volume pieces, every step adds faces, so the count can be as big as you like.
- The real lesson: 'half a cube' is ambiguous. Once you notice that, many answers become reasonable, and you start asking better questions: must the pieces match exactly? must the cut be flat?
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