One Punch, Two Folds
A square sheet of paper, 20 cm on a side, is folded twice before a single hole is punched through the packet. Fold 1: the top edge is brought down onto the bottom edge, creasing the sheet along a horizontal line through its centre. Fold 2: without unfolding, the left edge is brought over onto the right edge, creasing along a vertical line through its centre. The result is a 10 cm by 10 cm packet whose top and left edges are folded creases and whose bottom and right edges are free - four layers of paper. A hole is punched straight through the packet at a point 3 cm to the right of the left (folded) edge and 3 cm below the top (folded) edge. The sheet is then unfolded completely.
1. How many holes does the unfolded sheet have, and where are they? Show how each hole arises from a fold.
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Set coordinates. Put the sheet's corners at (0,0), (20,0), (20,20), (0,20). Fold 1 creases at y = 10; Fold 2 creases at x = 10. The packet is the square 10 ≤ x ≤ 20, 10 ≤ y ≤ 20, and the punch is 3 cm right of x = 10 and 3 cm below y = 10: point P = (13, 13).
Unfolding is reflecting. Undoing Fold 2 reflects P across x = 10, giving (7, 13). Undoing Fold 1 reflects every hole so far across y = 10, giving (13, 7) and (7, 7).
Four holes, at (13,13), (7,13), (13,7) and (7,7): a square of four holes centred on the sheet's centre, each 3 cm from both centre lines. Sanity check: four layers at the punch point, so four holes, arranged symmetrically left-right and top-bottom as the two folds require.
2. A second identical sheet is folded the same way, but this time the punch is made exactly on the left folded edge, 3 cm below the top folded edge. How many holes result, and where?
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Two holes, not four. The punch sits at (10, 13), which lies on the vertical crease. Reflecting a point on a mirror line returns the same point, so unfolding Fold 2 creates no new hole - the half-hole in each layer meets its mirror image and opens into one hole straddling the crease (a little longer along the crease than a free punch would be).
Unfolding Fold 1 then reflects (10, 13) across y = 10 to (10, 7).
Result: holes at (10, 13) and (10, 7) - both on the vertical centre line, 3 cm above and 3 cm below the horizontal centre line. A student who answers four has forgotten that a point on a mirror line is its own image.
3. Can a single punch through this two-fold packet ever produce exactly three holes? Explain using the possible sizes of the hole pattern.
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No. A single punch produces the orbit of the punch point under the two reflections, and the orbit can only be size 4, 2 or 1:
- 4 - the point is on neither crease (the general case, and the case in the first question).
- 2 - the point is on exactly one crease, so one reflection duplicates it and the other does not.
- 1 - the point is the sheet's centre (10, 10), which lies on both creases; every reflection leaves it fixed, so a punch at the exact centre gives one hole.
Three would need an orbit of size 3, and no combination of two perpendicular reflections can produce one. Three holes could only come from a different fold - for instance, a crease that misses the punch entirely - not from a different punch position.