Thursday, May 5, 2016

Cool Circuits: Hexagons

If you haven't read my previous post about Cool Circuits, better go read that first.

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:

Compared to the 90-degree pieces, these new ones are a little more stretched out.  If you've been paying attention you'll notice there's a new one.  The last piece in that image corresponds to the bit sequence 0001, which wasn't possible in the original Cool Circuits because it would have intersected itself.  I'm going to call it "Q".

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.

Behold!  Hexagons!
Because the turns are 60 degrees, the pegs on the board will be arranged in a hexagonal grid.  Clearly, the overall shape of the board should also be a hexagon, but what size?  Five seems like a nice diameter - it would have almost the same number of points as the original puzzle.  I modified my solver from the original puzzle for these new parameters, kicked it off, and sat back to watch the solutions roll in.  Sadly, there are none.

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.
Note how the V and N pieces can be swapped (and flipped and rotated) to convert one solution into another.  That's also true in some of the solutions to Cool Circuits, though in both cases it depends on how tightly-packed the circuit is.

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.

Time to raise the diameter to 7.  You can probably guess going into this one that there will be too much space on the board.  Going back to 8 pieces, I let the solver run overnight, and it says there are 1503 circuits.

Alternative boards.
So with lots of space (A) we have lots of solutions.  I also considered removing the six outer-most pegs (B: 474 solutions) and removing four pegs from each of three edges, forming a somewhat triangular board (C: 130 solutions), but the bigger hexagonal board seems more thematic to me.  The large number of solutions does mean that we'll need to have plenty of constraints in each puzzle to make it uniquely solvable.

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.

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