Problem 10 · 2015 Math Kangaroo
Medium
Spatial & Visual Reasoning
paper-cuttingfolding
A square bit of paper is folded along the dashed lines in some order and direction. One of the corners of the resulting small square is cut off. The piece of paper is then unfolded. How many holes are on the inner area of the piece of paper?
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Answer: B — 1
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Hint 1 of 2
The folds stack the 3×3 grid into one small square, so the single cut copies onto a matching point in every cell.
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Hint 2 of 2
A copied cut only becomes a hole when it lands strictly inside the unfolded sheet, not on its outer edge.
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Approach: track the cut through the folded layers
- Folding along the dashed thirds stacks all nine cells of the 3×3 grid into the small square, so cutting one corner of the stack puts a matching cut at the same corner of every cell.
- When unfolded, those copied cuts that land on the sheet's outer border only notch the edge, while a cut at an interior grid point makes a real hole.
- Exactly one of the copies falls on an interior point of the sheet, leaving a single hole, so the answer is 1 (B).
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