Paper folding and cutting for AFCAT: count the layers, then unfold in reverse

Paper folding questions show a square sheet being folded once, twice or three times along dotted lines, with arrows for the direction of each fold. Then a hole is punched or a piece is cut, and you choose how the sheet looks when fully opened. Nothing is hidden in these questions. Every fact you need is in the drawing, and two ideas do all the work.

Many candidates try to picture the whole unfolding at once and lose track after the second fold. The reliable way is slower to describe and faster to do: count the layers, then open one fold at a time.

Idea 1: layers

Every fold doubles the number of layers. A single punch through all the layers makes one hole in each layer.

Use this as a first filter. After two folds, any option without exactly four holes is wrong, as long as the punch is not on a fold line. That often removes two options before you think about positions.

Idea 2: reflection across each fold

When you open a fold, every hole on the folded part appears again as its mirror image across the fold line. Holes near the fold end up close together; holes far from the fold end up far apart.

So open the last fold first, reflect every hole across that line, then open the fold before it and reflect everything again.

How the examples are described

The sheet is a square split into a 4×4 grid of cells. Rows run 1 to 4 from top to bottom, columns 1 to 4 from left to right, and (2, 3) means row 2, column 3. In the result tables, ● marks a hole and · marks no hole.

The halfway lines are between rows 2 and 3 (horizontal) and between columns 2 and 3 (vertical). Under a fold along the horizontal halfway line, row 1 swaps with row 4 and row 2 with row 3. Under a fold along the vertical halfway line, column 1 swaps with column 4 and column 2 with column 3.

Worked examples

Example 1: one fold. The top half is folded down onto the bottom half. The folded sheet now covers rows 3 and 4. A hole is punched at (3, 1).

• Opening the fold reflects (3, 1) across the horizontal line: row 3 becomes row 2, so (2, 1).
• Two holes, at (2, 1) and (3, 1), one above the other on either side of the crease.

Example 2: two folds, punch at the outer corner. The left half is folded onto the right half. Then the top half is folded down onto the bottom half. The packet is now the bottom-right quarter (rows 3–4, columns 3–4), four layers thick. A hole is punched at (4, 4).

• Open the last fold (top–bottom): (4, 4) and (1, 4).
• Open the first fold (left–right): add (4, 1) and (1, 1).
• Four holes, one in each corner of the sheet.

Example 3: same folds, punch near the centre. Everything as in Example 2, but the punch is at (3, 3).

• Top–bottom: (3, 3) and (2, 3).
• Left–right: add (3, 2) and (2, 2).
• Four holes clustered round the centre.

Compare Examples 2 and 3. The same folds and a single punch give four holes each time, but in completely different places. Which corner of the packet you punch decides the pattern: the packet's corner at the sheet's centre gives a central cluster, and its outer corner gives the sheet's four corners.

Example 4: a diagonal fold. The sheet is folded along the diagonal from the top-left corner to the bottom-right corner, with the bottom-left half laid over the top-right half. A hole is punched at (1, 3).

• A fold along this diagonal swaps rows and columns: (r, c) becomes (c, r).
• Two holes, at (1, 3) and (3, 1).
• A punch at (2, 2), which lies on the diagonal, would give a single hole on the crease.

Cuts instead of punches

Cuts follow the same two ideas. A cut through a folded edge opens into a shape that is symmetric about the crease.

A cut on the open edges, where the separate layers lie loose, produces one piece per layer instead. Those pieces land on the sheet's outer edges.

Example 5. A sheet is folded into quarters as in Example 2. A quarter-circle is cut from the packet's top-left corner, the point where both creases meet at the centre of the sheet.

• That corner is the centre of the original sheet, and all four layers meet there.
• One full circle in the middle of the sheet.

A step-by-step routine

1. Count the folds and write the number of layers.
2. Mark the punch or cut on the final packet.
3. Open the last fold. Reflect across it.
4. Open each earlier fold in turn, reflecting everything you have so far.
5. Count your holes and check they match the layer count. If a punch sat on a crease, expect fewer.
6. Match the pattern to the options, checking position as well as count.

Practice set

Use the 4×4 grid described above.

1. A sheet is folded in half three times and punched once, away from any crease. How many holes appear when it is opened?
2. The left half is folded onto the right half, and a hole is punched at (2, 3). Where are the holes when the sheet is opened?
3. The top half is folded down onto the bottom half, then the left half onto the right half. A hole is punched at (4, 3). List the holes when the sheet is opened.
4. The sheet is folded along the diagonal from the top-left corner to the bottom-right corner, and a hole is punched at (2, 2). How many holes appear?
5. The sheet is folded into quarters as in Example 2. A small V-notch is cut in the packet's top-right corner, where the horizontal crease meets the right edge of the sheet. Where are the notches when the sheet is opened?
6. The sheet is folded as in Example 2, and a hole is punched at (3, 4). List the holes.

Answers:

1. 8. Three folds give 8 layers.
2. (2, 2) and (2, 3). They are side by side, close to the vertical crease.
3. (1, 2), (1, 3), (4, 2) and (4, 3). Two near the top edge and two near the bottom edge, in the middle columns.
4. One. (2, 2) lies on the diagonal crease, so the two layers are cut in the same place.
5. In the middle of the left and right edges. The top-right corner of the packet sits on the right edge at the horizontal crease. Opening the horizontal crease turns the cut into one triangular notch in the right edge, symmetric about the crease. Opening the vertical crease copies it to the left edge.
6. (2, 1), (2, 4), (3, 1) and (3, 4). Top–bottom gives (3, 4) and (2, 4). Left–right adds (3, 1) and (2, 1).

What to do next

• Fold a real sheet of paper into quarters, punch it with a pen, and check Examples 2 and 3 with your own hands. Five minutes of this beats an hour of staring at diagrams.
• Solve 10 folding questions a day for a week, writing the layer count first every time.
• Revise mirror and water images, since unfolding is repeated reflection, and move on to embedded figures.