Mirror and Transpose Traps

Three wrong answers made out of the right one

The generator does not invent its wrong answers. For every item it takes your correct pattern and reflects it three ways — left to right, top to bottom, and across the main diagonal — then offers the results as options alongside the truth.

That is why these options are so hard to reject on sight. A reflection of the correct answer has the correct number of holes, the correct spacing, and the correct overall look. Every hole is at the wrong address, and nothing else about it is wrong at all.

What each reflection does

The left–right mirror reflects the pattern across the vertical midline: a hole in column 1 moves to column 4, column 2 to column 3. Rows never change.
The top–bottom mirror reflects across the horizontal midline: row 1 moves to row 4, row 2 to row 3. Columns never change.
The transpose reflects across the main diagonal, the one running from the top-left corner to the bottom-right. A hole's row and column swap places, so row 1 column 3 becomes row 3 column 1. A vertical run of holes becomes a horizontal one.

The trap toggler

The correct answer below is three holes down the left edge, computed by the fold engine from a real fold-and-punch sequence. Each button applies one of the generator's own mutation functions to it and shows you what the resulting option looks like. Watch which holes move and which stay exactly where they were.

Showing the left–right mirror.

Correct answer

Left–right mirror

Correct answer beside the left–right mirror, produced by the generator's own mirrorH function — the same call that builds this option on a real item.

The whole run jumps from column 1 to column 4. This is the easiest reflection to catch, because every hole moves and moves far.

Counting holes cannot save you here

All three reflections are one-to-one maps of the sixteen positions onto themselves. The left–right mirror sends column c to column 3−c, the top–bottom mirror sends row r to row 3−r, and the transpose sends position (r, c) to (c, r). None of them can create a hole or destroy one.

A reflection trap always carries exactly the correct hole count, so no count check can ever eliminate one. Across roughly 18,000 generated wrong answers, every reflection option matches the correct answer's hole count exactly. The count check is a fine tool and the next two topics lean on it hard, but against these three families it returns "possible" every time. Position is the only discriminator you have.

When a mirror trap cannot exist at all

Before an option is accepted, the generator compares its signature against every pattern already chosen — starting with the correct answer itself. A duplicate is thrown away.

So if your answer is already symmetric about an axis, its reflection across that axis is the answer, the signature matches, and the candidate is discarded. That trap never reaches the item. This is not a quirk to memorize; it is why the mirror families are smaller than you would expect, and why the transpose is the reflection you will actually meet.

Reflection	Share of wrong answers (easy / medium / hard)	Answers already symmetric that way (easy / medium / hard)
Left–right mirror	4.8% / 9.6% / 10.4%	41.9% / 28.7% / 27.9%
Top–bottom mirror	5.0% / 8.6% / 10.2%	45.2% / 29.7% / 30.0%
Transpose	10.0% / 11.5% / 13.8%	6.1% / 14.3% / 12.5%

Read the two columns together. Roughly two in five easy answers are left–right symmetric, which kills that trap outright on those items — so the family is small. Barely one answer in sixteen is symmetric across the diagonal, so the transpose is available almost always, and it is the single largest reflection family at every difficulty. The reflection you are most likely to be offered is the one that looks least like your answer.

Together the three reflections account for 19.8% of the wrong answers on easy items, 29.7% on medium and 34.4% on hard — the largest identified block on hard items. (On easy items a punch in the wrong place produces more of them; that family gets its own topic at the end of this lesson.)

The cure: one landmark, two coordinates

Comparing whole patterns by eye is what these traps are built to defeat. Compare one hole instead.

Pick a landmark hole from your own derived answer. Against the two mirrors, any hole will do: on a four-wide grid the left–right mirror sends column c to column 5−c and the top–bottom mirror sends row r to row 5−r, and neither equation has a whole-number solution — no position is left in place, so a mirror moves every hole you have. Only the transpose fixes anything: it leaves the four cells whose row and column numbers match, R1C1, R2C2, R3C3 and R4C4. Take your landmark off that main diagonal and all three reflections move it.
Say its address out loud, as two numbers. "Row 3, column 1." Not "lower left" — a mirror image is also lower-something, and vague language is what the trap eats.
Check that address in each option and reject on the first mismatch. A left–right mirror fails on the column, a top–bottom mirror fails on the row, a transpose fails on both unless your landmark happens to sit on the diagonal.

One landmark usually settles all three. If two options both contain your landmark, take a second landmark and repeat — you will not need a third.

Try it on a live item

Below is a real generated item. Derive the answer yourself, pick a landmark hole, and give its row and column before you look at the options. The item's own explanation tells you how many holes to expect, which — as you now know — will not narrow the field by even one reflection.

The paper is folded as shown, then punched through all layers. Which pattern shows the holes on the fully unfolded sheet?

2 folds, then 1 punch through 4 layers — 4 holes appear when unfolded. Unfold in reverse order: each fold mirrors its holes across the crease.

A real generated item, options unmarked. Derive the pattern, name one off-center hole by row and column, then check that address against each option.

Then drill it in the hole punching practice generator, where these three families are waiting on every item.

The takeaway

Three of the wrong answers you meet are your own answer reflected: left to right, top to bottom, or across the main diagonal with rows and columns swapped. All three carry the correct hole count, so counting holes can never rule them out — only checking positions can. A reflection is suppressed when the answer is already symmetric about that axis, which is common for the two mirrors and rare for the transpose, making the transpose the reflection you meet most. Name one off-center hole by row and column, and check that address option by option.

Mirror and Transpose Traps

Summary

Practice Questions

AI Tutor