Thanks, Charles, for putting " the mystery and misery of the Flexisquare puzzle
to rest. "
Re: your question about how many permutations and combinations:
Focusing on the center square, if you blintz the outer corners behind,
you get a flat square described by U//X,
where U=BCEH and X=2358 specify the top and bottom faces.
There are (at least) 16 more such (small) squares with descriptions U//X
(different X).
Similarly for the other side V//X, V=1467. So here are 34 permutations and
combinations.
Of course, you described several more. I have no idea how many more??
You have perhaps rediscovered the "'uncut' silver even edge flexagon with a
flat main position link"
from p.267 of "Serious Fun With Flexagons" by Les Pook (2009).
Have a great day!
SVBE (si vales, bene est)
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Cheers, Ralph Jones
