Problem 9 · 2012 Math Kangaroo
Easy
Spatial & Visual Reasoning
paper-cuttingsymmetry
Werner folds a piece of paper once in the middle as shown. With a pair of scissors he makes two straight cuts into the folded paper, then unfolds it again. Which of the following shapes is not possible for the piece of paper to show afterwards?
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Answer: D
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Hint 1 of 3
After unfolding, the cut-out pattern must be mirror-symmetric about the fold line.
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Hint 2 of 3
Each straight cut on the doubled paper unfolds into a symmetric pair, so two cuts can make only a limited number of corners.
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Hint 3 of 3
Count how many separate notches or corners each shape needs and compare it to what just two straight cuts can produce.
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Approach: count the cuts each shape needs
- Folding once makes two layers, so each straight cut goes through both layers and unfolds into a pair of cuts that are mirror images across the fold.
- Two straight cuts can therefore create at most two such symmetric features — enough to make the single notch of A, the central hole of B, the trimmed corners of C, and the symmetric notch of E.
- Shape D has several separate zig-zag notches, more than two straight cuts can produce, so it is the one that is not possible.
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