
A puzzle (a hand game)
Posted: Posted December 29th, 2018 by Xhin

So, a bit of backstory. Over christmas my sister and I invented a game where we tried to get all the fingers of a hand out. Pointing to a finger would toggle some number of other fingers, and you were supposed to figure out the pattern.
After christmas, I programmed a version of this where the rules were randomized (with a few obvious constraints) so you had about a million different puzzles you could try:
http://gtx0.com/projects/handgame/
You can also try specific puzzles by sharing that ?seed link.
The basic way the puzzles are generated works like this:
* Each finger can toggle between 14 fingers (toggling makes up fingers go down and down fingers go up)
* The script makes sure that any particular finger can be toggled by at least one other finger, but never by all five fingers.
The actual math puzzle here
Is there a mathematical way of figuring out which set of finger rules are solvable and which are not? I have a way of doing a deep search to figure it out manually, but that's not really practical when setting up new games randomly unless I go through all ~2^25 puzzles individually.
If it helps, the actual ?seed is set up like this:
* The seed is a decimal representation of a 5bit binary number, ex  636580 = 10011011011010100100.
* This can be broken down into five chunks: 10011 , 01101 , 10101 , 00100
* Which then indicates which fingers each finger triggers (for example finger #2 triggers fingers 2, 3 and 5).


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