State diagram for penta-hexaflexagon

People have been exploring the hexaflexagon using the pinch flex and the Tuckerman traverse since the 1930's (see the Wikipedia article for more information). Bruce McLean first published an article on the v-flex in 1979 with a more recent article mapping the 9 sided hexaflexagon using the pinch flex and v-flex.

Below is a diagram mapping out the states of the 5 sided hexaflexagon using the pinch flex, v-flex, tuck flex and pyramid shuffle. You can create this flexagon using the color template described on the hexaflexagon page.

State diagram for penta-hexaflexagon

The above diagram shows which flexes to use to travel between the various states. The directional arrows indicate a tuck flex in one direction and an inverse tuck going in the opposite direction. The states are numbered 1-34. A single quote after the number indicates the state is the complement of another state. Swap 2's and 5's and then 3's and 4's to get the complement of a state. For example, the complement of a side numbered 123451 is 154321. A complement behaves the same as the original state, so if you can pinch flex from state 1 to 2, you can pinch flex from state 1' to 2'. States 1-34 and their complements total 65 unique states, since 1, 16 and 34 are self complements.

States 1, 2, 2', 3, 3', 4 and 4' have solid colors on both sides. You can use the Tuckerman traverse to travel between those sides. States 5, 6, 7 and 5', 6', 7' have four of one color and two of another on both sides. A single v, tuck or shuffle from the primary states take you to these secondary states. You can pinch flex between 5, 6 and 7 or 5', 6' and 7'. After that, the sides get increasingly mixed up.

The diagram below provides details on these states and where to perform a flex to take you to a different state. The two sets of six numbers for a state represent the two sides of the flexagon, with the top side going clockwise and the bottom side counterclockwise (hence each bottom number is directly opposite the corresponding top number). A flex line is drawn between two numbers to represent that the flex should use the vertex in between those two numbers as the current vertex for the flex. See flex notation for more information. Note that some vertices allow multiple flexes. For simplicitly, pinch flexes are not listed. Only states 1-7 permit the pinch flex.

flex guide for penta-hexaflexagon

A state is considered to be the same even after rotating or flipping the flexagon over. Also note that the leaves are not uniquely labeled, thus there are six copies of each of the numbers 1-5 distributed throughout the flexagon. If each leaf were uniquely labeled, every state except 1, 2, 2', 3, 3', 4 and 4' would come in three variations for a total of 181 states (58 * 3 + 7). This labeling would also demonstrate that states 17, 29 and 33 are actually different, though they look the same with the numbering I've chosen.

There are lots of interesting patterns to note here and ways to analyze how the various flexes interact. For example, you can see that 64 of the 65 states permit a tuck or inverse tuck. The pyramid shuffle can be performed in 39 of the states, the v flex in 18 and the pinch flex in only 13. Furthermore, you'll find that the slot flex can only be done in 10 of the states.

If you limit yourself to the pinch flex and v-flex, you can only get to 25 of the 65 states, as shown below.

pinch flex and v flex states

If you only use the tuck flex and pyramid shuffle, you'll find there are four disconnected sets of states, (3, 6, 7', 2'), (3', 6', 7, 2), 34 and everything else. This is shown below.

tuck flex and pyramid shuffle states

To figure out which flexes are required in order to visit all states, look at the full state diagram at the top. Clearly the tuck flex is required, because otherwise you couldn't get to states 26 and 27, among others. The inverse tuck flex is required or state 28 wouldn't be accessible. Likewise the pyramid shuffle is needed to reach states 27, 28 and 29. When you look at the diagram showing just the tuck flex and pyramid shuffle, you can see that all you need now is to be able to travel between the four clusters (3, 6, 7', 2'), (3', 6', 7, 2), 34 and everything else. With the exception of state 34, you could travel between these clusters using either the pinch flex or v-flex. Thus you can access 64 of the 65 states using the tuck and pyramid shuffle combined with either the pinch flex or v-flex.

The Missing State

So what about the final inaccessible state 34? You can't get to it using any of the flexes discussed so far, but it turns out there are multiple ways to reach it. One technique is shown below.

The video starts with state 2, uses a tuck flex to get to state 7, then a v-flex to get to state 13. From there, it demonstrates the inverse ticket flex, a combination of an inverse tuck, a couple pocket flexes and a tuck flex. This puts you at state 34. It then does the normal ticket flex from the opposite side to put you at state 13'. A v-flex takes you to 7' and another tuck flex takes you to state 2', where you can pinch flex back to state 2, the starting point. An alternate route to state 34 is to perform the slot-tuck-bottom flex from state 6.


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