What if Cool Circuits was made with a hexagonal grid instead of a square one? More precisely, what if the turns in each piece were 60 degrees instead of 90?
We'll use the same pattern for generating the pieces: Each piece is a path composed of five turns either clockwise or counterclockwise, and each piece is uniquely described by a sequence of four bits, where each bit indicates whether the next turn is in a different direction than the previous turn. Here's what the pieces would look like:
Can we make a circuits with all of these pieces? Nope. In 90-degree-land, each piece contributed a single 90-degree turn (either way) to the overall path, and since there were an even number of pieces that meant we could end up with the complete 360-degree turn necessary for a circuit. But with these new pieces, 6 of them contribute a 60-degree turn, and 3 of them contribute a 180-degree turn. That means if we use all the pieces we'll always end up 180 degrees off. Here's a tabular summary:
So we'll have to drop at least one piece. For now, just to be true to the original puzzle, we're going to have to let Q go. Sorry Q. Maybe we'll find a job for you later.
Cool Circuits has a neat property that every solution leaves behind 6 untouched pegs. In "Cool Circuits, Jr" they add 6 rings that you can use to wrap those pegs. Once you've done that, there is no position left untouched. This hexagonal version doesn't have that feature. For each turn you make, you make it impossible for the path to later hit the point you didn't turn towards. Near the edges, that can cascade and make other points untouchable as well. I suspect this is a clue to why there aren't any solutions in the 5-board.
Before we give up on this board though, what if we had even fewer pieces? I tried dropping J and E (leaving us with only pieces that contribute 60 degree turns), and I found that there are solutions to that variant. Exactly four of them (ignoring solutions that are equivalent via rotation and reflection). Here they are:
|Four unique solutions for the 5-board, using the UVNILS pieces.|
To me, this doesn't seem like enough solutions for an interesting puzzle game. Sure, with rotation and reflection we could make more circuits, but they would all end up feeling the same.
Overall, while not as elegant, I think this configuration works pretty well as a variant of Cool Circuits. I'm working on a web-based playable version called HEXFOLD, which I'll post here once I've finished the levels and had some people play-test it.