Problem 2 · 2023 AMC 8
Medium
Geometry & Measurement
spatial-reasoningsymmetryfolding
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Answer: E — It matches figure (E).
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Hint 1 of 2
Find the one corner where all the folds meet — that's the spot the cut affects most. Where does it land on the full sheet?
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Hint 2 of 2
Folding twice into quarters stacks four layers at one corner, and that corner is the center of the original sheet. So one cut becomes four identical snips arranged symmetrically around the middle.
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Approach: track where one cut lands once the layers are stacked
- The key question for any folding-and-cutting puzzle: where is the corner that all the folds meet at? Folding twice into quarters stacks all four corners onto one spot — and that spot is the center of the original sheet.
- So a single cut near that folded corner is really happening through four layers at once — meaning four identical snips appear around the center when you unfold.
- A straight diagonal cut takes a triangle off that stacked corner; unfolded, the four triangles join into one diamond-shaped hole in the middle — figure (E). This transfers: whatever you cut at the all-folds-meet corner becomes a symmetric shape centered on the sheet; cuts at an open edge stay at the edge.
Another way — fold real paper (or imagine one layer):
- If you can, fold a square twice and cut — the fastest check. If not, just unfold one step at a time: the cut on the folded square mirrors across the first crease, then the result mirrors across the second crease.
- Two mirrorings of a corner-triangle produce a four-fold symmetric hole sitting dead-center — matching (E).
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