Gifts · paper · state graphs

The Face Inside the Fold

A hexahexaflexagon has six faces. It has nine states, because three of those faces each turn up in two of them, showing the same six numbered triangles in the same order and allowing different flexes. Fold one here, or print the strip and fold it in your hands. Either way the page draws the graph the paper is actually walking on.

A hexahexaflexagon is a strip of nineteen paper triangles folded into a hexagon. Pinch three alternate corners together, open it from the middle, and a face appears that was not there a moment ago. It has six faces. The name says six, the instructions say six, and after twenty minutes most people have seen all six and put it down.

Six faces is right. Six states is wrong, and it is wrong by three. This page folds one in front of you from a written specification of the flex, derives the whole state graph, and then hands you the case that settles it: two positions that show the same face, with the same six numbered triangles in the same order, from which different flexes are possible.

Flex it

The hexagon below is a real hexahexaflexagon in pat notation, folded from the strip printed further down. Each triangle carries its face number in large type and, in small type, which of the nineteen strip triangles it is. Pinch at either set of alternate corners.

The flexagon

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Two things are worth noticing before anything else. First, one of the two pinches is often refused: at six of the nine states only one of the alternate corner sets will open at all, which is why the paper version feels stuck so often. Second, the button marked Other state, same face is not always available, and when it is, the hexagon it gives you shows the same face, with the same numbered triangles in the same order, and a different set of flexes.

The strip, at true size

The screen version is a simulation of a specification. The paper version is the thing itself, and the two are meant to disagree if either is wrong. Print the sheet, fold it, and tell the page what you are looking at.

The sheet carries four registration squares and a printed 150 mm ruler, from /_kit/printable.js, plus a 100 mm vertical scale of this page's own. Printers rescale by default and never announce it: fitting an A4 sheet onto US Letter, all by itself, costs 94.1%. That does not change a single number on this page: the state graph of a folded strip is a fact about the folding and not about its size. What it does change is how big your flexagon comes out, and whether your printer stretched the two axes by different amounts, which would leave the triangles not equilateral and the hexagon refusing to close.

Not calibrated. The page will not tell you the finished size until you have measured the printed ruler.

Folding it

  1. Cut the strip out along its outline. Crease every internal line both ways, then flatten it again.
  2. First fold. Turn the strip over to the side numbered 4 4 5 5 6 6. Fold along the strip so each pair of equal numbers meets face to face: the two 4s together, the two 5s together, the two 6s together, all the way along. You are left with a straight strip of nine triangles showing only 1s, 2s and 3s.
  3. Second fold. Fold that nine-triangle strip exactly as you would a trihexaflexagon: fold it back on itself at every third crease, bringing like numbers face to face, three times. It curls into a hexagon with one triangle left over.
  4. Glue the blank tab behind the first triangle.
  5. Read the number on each side and enter both below. Eighteen ordered pairs are possible out of thirty: the nine states of the map, and the same nine held the other way up, which is which face you happen to have put upwards on the table. If yours is not one of the eighteen, the fold is wrong, and the page will say so rather than guess.

Waiting for a pair.

Six faces are not six states

Here is the sentence this page exists to answer, in the voice of somebody who has just folded one and flexed it for ten minutes:

It has six faces. When I have seen all six I have seen everything it does, and the only question left is how quickly I can get round them.

That sentence contains a model, and the model is that the flexagon is a machine whose state is the face you are looking at: six states, and what you can do next depends only on which face is up. It is a reasonable model. Every hexaflexagon page on the internet is written as though it were true, and for the three-faced one it is true.

It is false here, and the case that breaks it is not an argument. It is a pair of positions. Two of the nine states show the same face, with the same six numbered triangles in the same order round the hexagon. From one of them, a face is one flex away that is not one flex away from the other. The face is the same and the next move is not, which is the one thing a machine with one state per face cannot do.

The discriminating pair

A machine with six states, one per face, cannot do that. Given the face, its next move is determined; here the face is given and the next move is not. The extra information is real and it is physical: the two positions differ in how the eighteen leaves are piled under the six visible ones, two and four deep in one of them and one and five deep in the other. Turn the flexagon over and the face underneath differs too. So the hidden state is not invisible, it is merely not on the side you were looking at. What is not available at all, from either side, is a six-state model that fits.

What this page means by "the same", and what it cannot see. The test above is the one this page can compute: which of the eighteen numbered strip triangles are facing you, and in what cyclic order. It does not see which corner of each triangle is pointing at the middle of the hexagon, and that is a real, visible thing. Gardner's own figure marks it, with a wedge in one corner of every leaf and a Y in another; the published count says there are eighteen configurations of the six faces on that reading and that only fifteen of them can be flexed, the three missing ones being put down to the 4, 5 and 6 tiles at the back flap. That is three ways each for faces 1, 2 and 3 and two each for 4, 5 and 6. Counted our way there are twelve, which is two ways each for all six. So we agree about 4, 5 and 6 and we are one short at 1, 2 and 3, which means at least one of the two pairs we call identical at each of those faces is a pair a marked strip would tell apart. We cannot tell you which, because the corner that comes to the centre is not represented anywhere in this model.

The table also prints the pile depths, which differ between the two, and which you can count at the rim of a paper one: that is another real difference, and it is not part of the test either. If the two positions look different in your hands, that is an attribute we are not modelling and not a misfold. What does not depend on any of it is the claim being made: the same face is up, the same six numbered triangles are facing you in the same order, and the flexes available are different.

The map, and the tour that walks all of it

Nine states, and every flex from every state, is one half of the object. Drawn out it is a triangle of three states with a triangular ear hanging from each corner. The three ears are where the hard faces live. The other half is what you get by turning the paper over: the same nine states with top and underneath swapped, eighteen positions in all, and no pinch flex ever crosses between the two halves. Everything counted below is counted on one half, which is why three faces turn up twice and not all six.

The state graph

Each node is labelled top/underneath. An arrow is one pinch flex. Nodes you have stood on are filled; flexes you have used are drawn bright.

Tuckerman's traverse is the old answer to how do I see all six: keep pinching at the same corner, and when it will not open, rotate the flexagon one corner and carry on. Walk it and it uses every one of the twelve flexes exactly once and returns you to where you started. The count is not six and it is not nine.

Conrad and Hartline's 1962 report gives that count for a flexagon of order N as 3N minus 6 flexes and N rotations. Both halves hold here, once you say what you mean by finished. The flex half is unconditional: twelve here, three on the three-faced one, from every starting position we tried. The rotation half depends on where your thumb is. Counting the tour from all 54 places you could start (nine states, six ways to be holding each), and stopping the moment every flex has been used and the starting state is showing again, it takes six rotations from 36 of them and five from the other 18, because the last rotation is sometimes not needed to finish. Keep going instead until the flexagon is back in exactly the position and the grip it started in, and all 54 give the same answer: three laps, 36 flexes and 18 rotations, which is twelve and six per lap. On the trihexaflexagon the same walk gives three laps of three flexes and three rotations, which is Conrad and Hartline's own worked case. The flex count is a fact about the object; the rotation count is a fact about your grip until you close the cycle, and then it is 3N minus 6 and N.

The three faces on one side of the paper strip are visited three times each; the three on the other side once each. That asymmetry is the reason the classic complaint about these things (faces 4, 5 and 6 are hard to find) is a fact about the graph rather than about your fingers. It also gives the anchor. Tuckerman's traverse has been in print since Martin Gardner's Scientific American column of December 1956, and the Wikipedia article on flexagons prints it as a twelve-symbol cycle of face numbers. That transcription is where we read it, uncited there, and the column itself we did not open. Nothing in the derivation above uses it. It enters once, in the check below, as the thing our own answer has to match.

The check

Everything above is computed here, now, by two implementations of one written rule that share no code, reconciled through /_kit/concur.js. Nothing on this page is a stored answer.

What is assumed, and what is derived

Assumed. One thing: the folded starting position, written in pat notation, which is the classical hexahexaflexagon. It is stated in the page's source as [[[3,-4],[1,-2]], [6,-5], [[9,-10],[7,-8]], [12,-11], [[15,-16],[13,-14]], [18,-17]], eighteen numbered paper triangles in six piles, and nothing else about the object is put in by hand.

Derived from it. That the thirty-six triangle sides fall into exactly six faces of six. That those faces lie along the strip as 1,2,3 repeating on one side and doubled pairs on the other, which is the classical printed template and is also, exactly, the pairing the first fold brings together. That there are nine states, twelve flexes, and one tour up to where you start. None of that was told to the program.

Not settled. The names of the six faces are a printing convention. Our tour and the published one are the same cyclic word under a renaming of the six faces: exactly three of the 720 possible renamings of six labels do it, and exactly three others turn our tour into that cycle read backwards. Three of each, because the map has a threefold symmetry that leaves our tour word alone; the split into forwards and backwards is a reflection, which is the statement that our traverse runs the map in the opposite sense to the one that was printed. Under our own numbering of the strip we could not reach the published literal digits by any choice of where to start numbering or which way to run. The structure is the same either way, and which of the two conventions is the printer's we could not settle.

Also not settled. Conrad and Hartline's formula, 3N minus 6 flexes and N rotations for a flexagon of order N, is confirmed here at the two orders we actually built, N = 3 and N = 6, on the full-cycle count. We did not build orders 4, 5, 7 or 8, so we do not assert the general formula. Their report is unrefereed and we read it in a 1998 conversion to HTML, not on paper.

Left open. The published count of how many ways a face can be shown is fifteen of a possible eighteen, and it counts which decorated corner of the leaves comes to the centre. This page counts twelve, because it counts which numbered triangles are facing you and in what order. Neither number is wrong for what it counts, and this page's model does not represent the corner at all, so it cannot tell you whether the pair it displays would survive that finer test. Folding one and looking is the way to settle it, and that is a reader's hands, not a check we can run.

What agreement does not buy. Both engines implement one written specification of the pinch flex. If that specification is misread the same way twice, this page cannot see it. What the printed strip is for is that a reader's hands are not a third implementation of the same misreading.

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