Paper Folding and Cutting

This is the core of the chapter, and every other type is built on it. The sheet is folded once, twice or three times, a small hole is punched through the folded paper, and the options show the opened sheet. Four smaller forms of the same idea live inside this type: counting the holes, the folded figure itself, the 'which cannot be the opened sheet' question and the reverse question. All of them use one fact: unfolding is a mirror image across the crease.

The logic. A punch goes through every layer. So every layer gets a hole. When the layers are opened, each hole lands at the mirror position of the hole in the layer it was folded onto. That is why we open the folds in reverse order and reflect all the holes we have so far.

The method.
1. Count. For full half-folds and holes away from the creases, holes = punches × 2^folds. Strike out every option with a different count.
2. Find the stationary part. The part of the sheet that never moved is exactly where the folded square sits. The punch you see is in that part, at the same place as drawn.
3. Undo the last fold first. Mirror every hole across that crease and keep the originals. Repeat for the earlier folds.
4. Check the symmetry (for half-folds about the centre lines). The opened sheet must look the same in a mirror placed on each such centre line. For any other fold, do not rely on this check; unfold in reverse order and reflect the marks at each stage.

[[fig:0]]

Reflection moves a hole the same distance to the other side. It never slides the hole by a fixed amount. A hole 1.5 above the crease has its copy 1.5 below, so a hole close to the crease has a copy close to the crease.

Counting form. When the question asks 'how many holes', work with numbers only. Layers and holes are different: after two folds there are 4 layers, but two punches make 8 holes. If the fold is not a half-fold (a corner fold, a fold into thirds), do not use 2ⁿ; count the layers lying under the punch. If the question asks for the holes in one half of the opened sheet, use symmetry: holes off the creases split equally between mirror halves.

[[fig:1]]

Fold-only form. Sometimes there is no punch and the answer is the folded sheet itself. It is the stationary half exactly as it was, together with the flipped copy of the half that moved. Check the outline first (a half-fold gives a rectangle of half the size), then send the most unusual part across the crease.

[[fig:2]]

'Which cannot be' form. The punch is not fixed here, so do not search for 'the' right option. Test each option with a mirror on each crease. The option in which a hole has no partner at the same distance on the other side is the answer.

[[fig:3]]

Reverse form. The opened sheet is given and you must find the folds. A crease must be a mirror line of the pattern, and 2, 4 or 8 identical holes mean 1, 2 or 3 folds. Four holes symmetric about both centre lines point to two half-folds; four holes symmetric about both diagonals point to two diagonal folds.

[[fig:4]]

Here the cut is not a small hole floating in the middle. It touches a crease, the outer edge of the paper, or the corner where two creases meet. This changes the count and the look of the answer, and it is the second most important type in the chapter.

The logic. A cut that touches a crease sits right next to its own mirror image. The two halves meet along the crease and open into one bigger shape. A cut that touches only an outer edge has nothing to join, so it stays open at the edge and makes a notch. A cut away from every side stays a separate shape and is copied as in the punch type.

[[fig:0]]

In the picture the folded sheet is on top and the opened sheet is below.

• A half-circle with its straight side on a crease becomes a full circle.
• A triangle at the crease corner of a twice-folded sheet becomes a diamond (a square turned by 45 degrees) at the centre of the sheet.
• A quarter-circle at that corner becomes a full circle at the centre.
• A right-angled triangle with one leg on a crease joins its mirror image into a bigger isosceles triangle; an isosceles triangle with its base on the crease would give a diamond.
• A cut on an outer edge makes a notch, copied to the edges that the folds brought together.

The method.
1. Read the four sides of the folded sheet. The sides towards the middle of the paper are creases; the other sides are original outer edges.
2. For every side of the cut, ask: is it on a crease, on an outer edge, or free?
3. Mirror the cut across the crease it touches first and draw the joined shape.
4. Copy the result across the other crease, last fold first, as in the punch type.

Counting the shapes. For a twice-folded sheet a free cut gives 4 copies, a cut touching one crease gives 2 joined shapes, and a cut at the crease corner gives 1. In general, the copies are 2 raised to the number of creases that the cut does not touch. A notch on an outer edge is not on a crease, so it is still doubled by each fold.

[[fig:1]]

Reading the answer backwards. If the opened sheet is given and the folded cut is asked, look only at the stationary quarter: the cut on the folded sheet is exactly what lies there. The cuts in the other quarters are flipped copies, so never slide one of them back.

[[fig:2]]

This is the form in which most questions of the chapter appear. The cut is not a circle, which looks the same after any reflection. It is a triangle, an arrow, an L or a letter such as F, a shape with a clear direction. The positions of the copies are easy; the whole question is the direction of each copy. A wrong option usually has every shape in the right place and one of them turned the wrong way.

The logic. A fold is a reflection, and a reflection has one simple effect on any shape:

• The part of the shape along the crease keeps its direction.
• The part of the shape across the crease (towards or away from it) reverses.

So a vertical crease flips the shape left to right and a horizontal crease flips it top to bottom. One reflection flips a shape; two reflections in perpendicular creases turn it by a half-turn.

[[fig:0]]

The picture shows an F punched once in a sheet folded twice. In the opened sheet the top-right F is the original. The top-left F is its mirror image. The bottom-right F is flipped upside down. The bottom-left F, where both reflections act, is the original turned by a half-turn (180 degrees).

[[fig:1]]

What each fold does to a symbol.

• Vertical crease: left and right swap. Horizontal crease: top and bottom swap.
• Two perpendicular creases: the copy in the opposite corner is a half-turn of the original, not a mirror image.
• A diagonal crease lays a vertical stem down into a horizontal one (see the diagonal type).
• A quarter-turn never comes from folding. An option showing a symbol turned by 90 degrees is a rotation trap.

The method.
1. Pick one distinctive part of the symbol, such as the long arm of the F or the right angle of the triangle.
2. Send that part across the last fold first, then across the earlier fold, and place the copies.
3. A neat check: a part that points towards a crease points towards it in the copy too, and a part that points away still points away.

[[fig:2]]

The last picture is the shape test. A copy that has the same direction as the original, a slid copy, can never come from a fold.

In a diagonal fold the crease runs from corner to corner, so the folded sheet is a right-angled triangle and the mirror line is slanted at 45 degrees. This is the one type where the square-grid habit of 'same height' or 'same column' stops working. Read the logic once and the method becomes simple.

The logic. A slanted mirror turns a horizontal distance into a vertical one and a vertical distance into a horizontal one. So the reflection swaps the two coordinates. For the diagonal from (0, 0) to (8, 8), the line y = x, the formula is (x, y) → (y, x). The hole at (6, 2) goes to (2, 6). For the other diagonal, x + y = 8, the formula is (x, y) → (8 − y, 8 − x). A point on the diagonal is its own mirror image and does not double.

[[fig:0]]

How many holes? A diagonal fold is one fold, so it gives 2 layers and one punch makes 2 holes, not 4. Two diagonal folds (the second along the symmetry line of the triangle) give 4 layers and 4 holes.

Mixed folds. The diagonal is often the last fold after one or two half-folds. Undo in reverse order as always: first the diagonal (swap), then the half-folds (8 − x or 8 − y). With a half-fold, a half-fold and a diagonal fold, one punch makes 8 holes in a ring around the centre.

[[fig:1]]

The method.
1. Count the holes: 2 for one diagonal fold, then multiply by 2 for every later fold you must undo.
2. Reflect one hole across the diagonal using the swap, last fold first.
3. Check the picture: the hole and its copy lie on a line at right angles to the diagonal, at equal distances from it.

If the opened pattern is a mirror image along a slanted line, the crease was a diagonal. The four holes of two diagonal folds are symmetric about both diagonals, not about the centre lines. This also separates a diagonal fold from a half-fold in the reverse question.