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Fold and punch

A square sheet is folded in half, then again, and punched through. Which picture shows the holes when it is opened out? More folds, punches and squares.

  • Spatial
  • Untimed
  • Endless levels

How it plays

The board is the sheet as it lies after folding: the dotted squares are where paper was folded away, and ● marks the punch, which goes through every layer. The folds are listed in the order they were made.

Open the folds last first. Each fold opens like a book, so every hole gets a twin mirrored across that fold line; after f folds a punch has become 2^f holes. The wrong pictures are the usual slips: a fold left shut, the holes slid across instead of mirrored, the wrong middle line, the punch misplaced.

Levels add folds (up to the most you set), then punches, then a bigger sheet, without end.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises the paper-folding items of spatial tests: holding a fold in mind and reflecting across it. It claims nothing about spatial ability beyond such items.

Learn

How to get better at fold and punch: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the folds in order, and find the folded sheet on the board: the plain squares are the stack that is left, the dotted ones were folded away, and ● is where the punch goes through every layer.
  2. Count the holes you expect: every fold doubles them, so each punch becomes 2^f holes after f folds. Rule out every picture with the wrong count.
  3. Open the folds last first. The last fold’s crease is an edge of the stack; reflect every hole across it, then reflect the lot across the fold before, and so on back to the first.
  4. Check the survivors against the two fixed facts below — the ● squares stay holes, and the picture is symmetric about the first fold — before you choose.

Techniques and tricks

  • Count first

    Holes = punches × 2^folds. The “fold left shut” picture always has half the holes, so a count throws it out without any folding.

    e.g. 2 punches, 3 folds: 2 × 8 = 16 holes

  • Same distance from the crease

    A fold opens like a book, so a hole and its twin sit the same distance from the crease, one each side. A hole touching the crease has a twin touching it on the other side.

    e.g. 4-wide sheet, left folded over right: a hole in column 3 gets its twin in column 2, a hole in column 4 one in column 1

  • The ● stays a hole

    Unfolding never moves the punch itself. The right picture has a hole on every ● square, and no other hole inside the stack. The “punch misplaced” picture fails this at once.

  • Symmetric about the first fold

    The first fold was made across the middle of the whole sheet and is the last one you open, so the finished sheet is a mirror image of itself about that middle line. A picture that is not cannot be right.

  • Upright crease, same row

    Folding left or right moves holes along their row; folding top or bottom moves them along their column. A twin in a different row after a left or right fold means the wrong middle line was used.

  • Track one hole, then copy

    With several folds, follow a single punch all the way out, writing down where its 2^f holes land. Each other punch makes the same pattern from its own square.

As the levels climb

  • A second fold comes in, then a second punch, then a bigger sheet. With two folds the stack is a quarter of the sheet; unfold one at a time rather than picturing both at once.
  • Then a third fold and a third punch. At three folds one punch is 8 holes, so lean on the count and the symmetry checks to clear the field before you fold anything.
  • After that the sheet grows two squares a side at a time, without end. The folds and punches stay the same, but there are more squares for the holes to hide among, so use the crease distances, not the picture’s overall look.
  • The number of folds stops at Most folds in the settings. Each axis takes at most two folds, so the sheet halves at most twice each way.

Traps to avoid

  • A fold left shut: the picture with half the holes. You opened all but one fold — count before you choose.
  • Holes slid across instead of mirrored: the right number of holes, but copied to the same place in the other half rather than reflected, so a hole near the crease lands near the far edge. Check crease distances.
  • The wrong middle line: the holes mirrored top to bottom when the fold was left over right, or the other way. Check that each twin shares a row (left or right fold) or a column (top or bottom fold) with its partner.
  • The punch misplaced by one square: the whole pattern shifted. The right picture has holes exactly on the ● squares.

How to practise

  • Set Most folds to 1 and play until you open a single fold without thinking, then raise it a step at a time to 4.
  • Raise Options shown towards 8 to see more of the slips at once — each wrong picture is one named mistake, and the working says which.
  • When you miss, read the working line by line: it gives the hole count after each fold is opened, so you can see the exact step that went wrong.
  • Fold and punch a real sheet once or twice with a hole punch or a pencil. Watching twins appear either side of a crease is what the mirror rule describes.

Levels

It is untimed unless you turn the clock on. There is no last level: at least one dial turns without end. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

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