Two Punches in a Twice-Folded Square
A square sheet of paper has its corners labeled P (top-left), Q (top-right), R (bottom-right), and S (bottom-left). The top edge is folded down onto the bottom edge, then the left edge is folded over onto the right edge, leaving a quarter-size square packet. The creases are sharp and the packet is not opened.
A hole is punched straight through the packet at a point one-third of the way in from the packet's left edge and one-third of the way down from its top edge. A second punch goes through the packet exactly at the midpoint of the packet's left edge - the folded edge created by the second fold.
The sheet is then opened flat and laid on the desk. Before anyone looks, the class must predict the total number of holes, where each one sits, and which corners of the original square are nearest to each hole. One student warns that the second punch will behave differently from the first because of where it sits.
1. Predict the total number of holes and describe the position of each one relative to the original square.
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The packet has four layers, so the interior punch is duplicated four times; the edge punch is not.
First punch (interior). It sits one-third in from the vertical center crease and one-third down from the horizontal center crease. Unfolding reflects that point across both center lines, giving four holes at the points one-third and two-thirds of the way across the sheet and one-third and two-thirds of the way down. They form a rectangle centered on the sheet, each hole nearest a different corner: the (one-third, one-third) hole nearest P, the (two-thirds, one-third) hole nearest Q, the (two-thirds, two-thirds) hole nearest R, the (one-third, two-thirds) hole nearest S.
Second punch (on the folded edge). The packet's left edge is the vertical center crease, and the midpoint of that edge is the sheet's center point, which lies on both creases.
Total: 4 + 1 = 5 holes - the four-hole rectangle plus one hole at the exact center.
2. Explain why the punch placed on the folded edge produces only one hole while the interior punch produces four, even though both pass through the same number of layers.
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Both punches do pass through four layers; the difference is what happens at the unfold.
A punch through the interior removes a full disk from each layer, and the four disks land at four different places, because the reflections across the center lines move the point.
A punch centered on the folded edge removes only a half-disk from each layer, since the other half of the punch circle hangs off the edge of the packet. When the fold is opened, that half-disk is reflected across the crease and joins its mirror image into one full disk. The same thing happens at the second crease: the center point of the sheet lies on both fold lines, so reflecting it across either line returns it to itself.
The general principle to carry away: holes multiply by the number of layers only when the punch is off every crease and every edge. A punch on a crease line halves the count along that fold, and a punch at the intersection of two creases collapses four positions into one.
3. Suppose the first punch were slid straight up until it sat exactly on the packet's top edge, while the second punch stayed where it was. How many holes would the sheet have, and where would they lie?
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The packet's top edge is the horizontal center crease, so the first punch now lies on a fold line but not on the vertical crease.
Applying the crease principle: reflection across the horizontal crease maps the punch onto itself, so that fold contributes no new holes; reflection across the vertical crease does move it, so the count is halved from four to two. The two holes lie on the horizontal center line, one at two-thirds of the way across the sheet and one at one-third, mirror images of each other.
The second punch is unchanged: one hole at the exact center, on both creases.
Total: 2 + 1 = 3 holes. All three sit on the horizontal center line, which is a quick sanity check - a viewer who predicts holes above or below that line has mislocated a crease.
Note the direction of change: moving a punch toward a crease reduces the hole count, so a student who expects "same punch strength, same number of holes" will overcount.