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What a Fold Does to Layers
The Hole-Count Law
Summary
Holes equal layers
Every item punches one position, and a punch takes every layer under it. So the correct answer's hole count is decided before you work out a single position: count the layers stacked at the punched cell, and that is how many holes the right grid has.
Getting that number is much cheaper than placing holes, and it is worth having first.
Two folds do not mean four holes
Fold the sheet in half, then in half again. Every position still under paper is exactly four deep:
Punch anywhere on that quarter and you get four holes. Two folds, four holes — which is where students pick up the rule "holes = 2 to the power of the number of folds."
Now make two folds of a different kind. A quarter fold, then another quarter fold:
Three deep in one column, one deep in the next, and the other half of the sheet empty. Punch the three-deep column:
Two folds. Three holes.
The powers-of-two rule is false. It happens to be right in the one case where every fold is a half fold, and it fails the moment any other fold appears — which on this section is most of the time. Odd hole counts are ordinary, not exotic. Trust the layer grid, never the fold count.
Only a half fold doubles the whole sheet
That is the whole mechanism. A half fold lays paper over everything that stays behind, so every covered position doubles at once. Any other fold moves a strip or a corner, doubling the region it lands on and leaving the region it never reached at whatever thickness it already had. A quarter fold leaves single-layer paper sitting right beside doubled paper, and that unevenness is what breaks the arithmetic.
Watch it survive a third fold. Two quarter folds and a half fold give this:
Six, not eight. Punching that column pierces six squares:
And the column beside it is only two deep, so the same paper after the same three folds yields two holes if the punch lands one position over. The count is a fact about the punched cell, not about the fold list.
What the count rules out
A grid with exactly one hole is always wrong. The punch always sits on at least two layers in a real item: if a punch would produce a single hole, the generator throws that item away and builds another. Across 1,800 generated items, not one correct answer showed a single hole. Wrong answers show them constantly — 416 single-hole grids appeared among the 7,200 options in that same sample. A lone hole is a free elimination.
The rest of the range is narrow. Correct answers in that sample carried 2, 3, 4, 5, 6 or 8 holes; never 7, never more than 8, and never more than 4 on an easy item. Anything outside that range is not a candidate.
As a first-pass filter the count earns its keep: measured against generated items, it eliminated about a third of the wrong answers on easy items and about half on medium and hard ones.
2 folds, then 1 punch through 3 layers — 3 holes appear when unfolded. Unfold in reverse order: each fold mirrors its holes across the crease.
Two folds, one punch through three layers, three holes — the item's own explanation says so. Two of the five grids carry a single hole and die immediately. Two more carry two holes. Exactly one grid carries three, and it is the answer. Counting alone finished this item; often it will not, and the surviving candidates still have to be separated by where their holes sit.
The takeaway
One punch, so the correct answer's hole count is the layer count at the punched cell. Only a half fold doubles the entire sheet; quarter folds thicken one strip and leave the rest thin, which is why counts of 3 and 5 are perfectly normal and why "2 to the power of the folds" is folklore. A correct answer never shows one hole and in practice carries 2 to 8, with easy items capped at 4. Count first: it costs almost nothing and it clears out roughly half the field.
Practice Questions
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