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Hard-Item Features
Diagonal Folds
Summary
Two shapes, twelve creases
Every diagonal crease in this section runs at 45 degrees and splits the squares it crosses corner to corner. There are only two shapes it can make: the sheet folded corner to corner into a big triangle, and a corner turned in as a flap.
The generator's entire diagonal vocabulary is twelve creases — four corner-to-corner folds (two diagonals, either half being the one that moves) and eight corner flaps (one at each of the four corners, in two sizes). Nothing else is legal, so there is no third shape waiting to surprise you.
They never appear on easy items. Easy folds are vertical and horizontal only; medium and hard both draw on the diagonals, and they are common at the top end — about three in five hard items finish with a diagonal edge in the folded outline.
The corner-to-corner fold
One crease, and half the square stops existing.
The paper that moves lands exactly on the paper that stays, so the surviving triangle is two layers thick everywhere. A punch there gives two holes: the punch position and its reflection across the diagonal.
Six cells at two layers, six cells empty, and the four cells the crease runs through still showing one. The reflection across a corner-to-corner crease is a swap: the row number and the column number trade places. The punch above sits at R4C2, so its partner is R2C4 — and those are the two holes on the answer grid.
The corner flap
The other shape turns a single corner in. Most of the sheet never moves.
Count what that flap actually did: one cell went to two layers, two cells sit on the crease, one corner cell emptied, and the remaining twelve are still single sheets. Thirteen cells can legally be punched and exactly one of them holds more than a single layer. The flap thickens a wedge; it does not remodel the sheet.
The same flap comes one size larger, cutting three cells off the corner instead of one:
Three cells doubled now, three sitting on the crease, three emptied and seven untouched — ten of the thirteen covered cells are still a single sheet. A larger wedge, and the same conclusion.
The reflection works the same way for a flap as for the full diagonal, just measured from a crease that does not run through the corners. The reliable way to run it: a 45-degree crease moves a hole along the other diagonal — one step up is one step across — as many steps past the crease as the hole sat before it. In the flap above, the punch at R3C2 pairs with R4C1: one step down, one step left.
The crease band is off limits
A diagonal crease cuts the cells it crosses in half. Part of such a cell lies under the fold and part does not, so a punch there would tear through a folded edge and unfold into partial holes. The engine refuses to place a punch on any cell a crease has cut, which is why the punch always sits off the crease band.
That has a consequence worth more than the rule itself. A cell that has been cut can never receive a hole either — across every legal three-fold sequence the generator can build, 24,357 punch outcomes in all, not one hole lands on a creased cell. So on a corner-to-corner item the diagonal band of the answer grid is guaranteed empty, and any option with a hole sitting on that diagonal is wrong before you check anything else. This is how the practice engine behaves; the official instructions describe the punch but do not spell out where it may sit.
Read the layer grids above with that in mind. They report layers, not permission: the cells showing one layer along the crease are drawn like any other covered cell, and they are still unpunchable.
A diagonal inside a fold sequence
Diagonals do not arrive alone. Here is one folded onto a sheet that has already been halved:
The vertical fold doubled the left half; the diagonal then folded the doubled sheet onto itself, and one cell reached four layers while its neighbors stayed at two. Punching the four-layer cell at R2C1 gives four holes: R1C2, R1C3, R2C1 and R2C4. Nothing about that answer is guessable from the outline — it comes out of the reverse walk, one crease at a time, exactly as an orthogonal item does.
Here is a real hard item with a diagonal in it:
3 folds, then 1 punch through 6 layers — 6 holes appear when unfolded. Unfold in reverse order: each fold mirrors its holes across the crease.
The takeaway
Diagonal creases come in two shapes and never at easy difficulty. Corner to corner empties half the square, doubles the other half, and reflects a hole by swapping its row and column. A corner flap doubles a small wedge and leaves most of the sheet single-layered. The cells a crease cuts are neither punchable nor ever holed, so the crease band on the answer grid stays empty — check it first and a whole option can fall for free.
Practice Questions
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