Simple foldability — the series
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The fold a machine can make
A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.
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The patient machine is the weak one
A machine that folds one layer at a time sounds like a machine with more freedom, not less. It has less, and the reason is the most ordinary fact about paper there is: it is joined, so whatever a machine declines to hold it also cannot move.
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A machine that can only crimp
Change the atom and the whole picture changes. A machine whose single move folds two adjacent creases at once reaches strips no simple-fold machine reaches, is defeated by strips they handle easily, and cannot fold an odd number of creases at all — for reasons that are pure arithmetic.
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The machine that may choose
Three restricted machines lose patterns that fold perfectly well. Give one of them a choice — any block of layers, top or bottom — and the loss vanishes: over a hundred and seventeen spacings, every flat folding of every strip became reachable. Being forced was the whole problem.
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Where the machine catches up
The weakest machine in the subject folds every layer at once and is stopped by a strip with two creases in it. On a strip of equal stamps it is stopped by almost nothing: every one of the 288 folded states a six-stamp strip has is reachable by a sequence of all-layers folds, and on every unevenly creased strip tried it reaches none of them. At seven stamps the completeness ends, and finding out where it ended is what checking it past six was for.
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Deciding is not making
Four earlier essays here ask which machines can flatten a strip at all, and the answer sorts them into a lattice with one column full and three with holes in it. Asked instead what each machine can produce, the three sort completely differently: the machine that may choose its block reaches every folded state of every strip tried, the machine that takes one layer reaches exactly four whatever the strip is and however long, and the machine that takes the whole pile is the only one whose answer depends on the spacing at all.
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The easiest strip needs the deepest reach
The patient machine and the machine that may choose are the two ends of one number: how many layers of the pile a machine is allowed to hold. At one it reaches four states whatever the strip; at the pile's full depth it reaches everything. In between it is a machine nobody has defined, and measuring where completeness arrives inverts these essays' own ordering — the evenly creased strip, which the machine that takes everything folds perfectly, needs the deepest reach of all, and one uneven strip is complete at two.
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A shallow machine pays in states, not folds
A machine allowed to take only a few layers of the pile at a time reaches fewer folded states, and the natural fear is that it also reaches the ones it does by much longer sequences. Walked breadth first, so that every state's shortest sequence is found, it does not. On unevenly creased strips every state takes exactly one fold per crease at every depth, because no two creases ever lie on one line. On strips of equal stamps a shallower machine needs one fold more for a minority of states and two more for eight of the 924 states at seven stamps — and never more than the crease count, which no machine can exceed.
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Fourteen states are one pile
A machine that folds every layer at once reaches every folded state of a strip of six equal stamps and misses fourteen piles at seven. The fourteen are not fourteen things. Taking a pile's bottom stamp and putting it on top maps foldings to foldings, so the 462 piles of seven stamps fall into 33 classes of exactly fourteen, and the missed piles are one whole class: the pile 0 6 1 2 3 4 5 — an accordion of five stamps with the last stamp wrapped round it and slid into the fold that holds the first — seen from each of its seven stamps. At eight stamps the machine misses 64 piles, and they are exactly the piles that leave that one when an end stamp is removed.
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Nine stamps have piles of their own
A machine that folds every layer at once reaches every folding of six equal stamps, misses one class of fourteen piles at seven, and at eight misses exactly the piles that shrink to that class when an end stamp is taken off. The prediction was that nine would do the same. Counted directly, nine equal stamps have 4,536 foldings and the machine reaches 4,052. Of the 484 it misses, 324 are inherited from eight and 160 are new — among them the nine-stamp tucked pile. And the symmetry that made the misses whole classes breaks: at nine stamps, turning a pile's bottom stamp to the top takes 52 piles the machine reaches to piles it cannot.
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The machine's misses have no short list
A machine that folds every layer at once misses fourteen piles of seven equal stamps, sixty-four of eight and 484 of nine, and the question left at nine was whether its failures could be described by a short list of forbidden piles. Counted to twelve stamps, and to thirteen by the same method, they cannot. What the machine reaches is closed under taking an end stamp away, for a reason that fits in a sentence, so every miss contains a smallest one, and the smallest ones keep arriving: 14, 0, 160, 12, 1,194, 116 and 7,786 from seven stamps to thirteen, the tucked pile among them at every odd length. Meanwhile the machine's reach grows by just under three a stamp and the foldings by more, so the share it can make falls from all of them at six stamps to under two thirds at thirteen.
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An odd strip can hook its own ends
A machine that folds every layer at once has smallest misses by the thousand on strips of an odd number of equal stamps and by the hundred on even ones, and the alternation had no explanation. Most of it is one arrangement. On an odd strip each end stamp's free edge faces the crease at the other end, so an end can lie inside the far end's fold with its edge pointing in: a hook. Nine in ten of the odd strips' smallest misses carry one, an even strip cannot carry one at all, and the misses without a hook — 12, 12, 82, 116, 442 and 740 from nine stamps to fourteen — stop jumping by a factor of a hundred between neighbouring lengths.