I’ll note that my coloring game doesn’t really depend on using a hexagonal tiling; I just wanted a a tiling with at most three tiles meeting at any point, and that was the simplest example.
You could just as well play the same game on something more complex like this, with the same rules (+$1 for points with clockwise RGB; -$1 for points clockwise RBG). I can tell you immediately on sight that the provided coloring breaks precisely even, and that you can do a lot better; given a minute to count, I can tell you that you can earn a maximum of $31 on that board.
You can even play the game on a tetrahedron or a cube instead (coloring the faces and gaining/losing dollars from vertices by the same rules as ever). Or a soccer ball! Indeed, you may find playing around with such figures particularly enlightening.
[SPOILER]a cryptogram existed which decodes into TWO different valid sentences. For some reason the Nevermore line popped into my head as one to try. I couldn’t construct a twin sentence, obviously, but the five components (bayou etc.) are all in a dictionary!
It was only later that I realized Nevermore was especially appropriate: Poe popularized cryptograms in his short story The Gold Bug.[/SPOILER]
Here’s a familiar type of puzzle (though this particular one has never been solved AFAIK until today.) Billy did his long division problem properly, but his sister, now part of a religious cult, needed to smudge out all the digits in the homework exercise except the single 2. Can you reconstruct the problem and its solution?
I should perhaps give a hint for my coloring puzzle, since I worry people must be giving up on it. My hint is this: go, pick a nice symmetric board, color it however you like, and calculate your profit. Then,
Change the color of the centermost tile any which way, and recalculate your profit
I guessed the basic premise on my own, but even if I hadn’t, a Google of those terms turns up the relevant Wiki page as the second hit (after this very thread).
Indistinguishable, I’m curious about your puzzle, but just haven’t had a chance to play around with it yet. I’m guessing, though, that you could start with an arbitrary initial state and allow it to evolve towards optimal by running something like the Demon cellular automaton.
I’ll summarize my conclusions about Indistinguishable’s puzzle, though I’m embarassed not to be anywhere remotely near a formal proof.
Especially embarrassing is that I noticed what became his Hint while I was playing around, but didn’t consciously proceed to an Aha! :o
[SPOILER] Paint Red-Green-Blue-Red-Green-Blue-… around the edges, easily taking $1 profit for every three edge tiles. For simplicity, completely avoid Red in the interior so you needn’t worry about any scoring there.
For the hexagon grids, this is easy. For the more complicated grid, there are 64 edge tiles (not counting two useless tits in the bottom corners) for a score of 64/3 = $21. When I colored that grid the details got complicated: some of the edge tiles are adjacent to 4 (or more) other edge tiles instead of just two. In some of those cases I scored an extra $1 on the edge, but had to pay back $1 to lock out a Red which was protruding into the interior.
This gave me enough satisfaction that I can quit this puzzle and move on with the rest of my life, though it’s a minisculely tiny sliver of satisfaction compared with Indistinguishable’s, who presumably has proof outlines for this result.[/SPOILER]
Good work, septimus! Your conclusions are correct.
On a flat board with no holes, the only thing that matters is how the outermost tiles are colored: run around the outside border in clockwise order calculating each red->green, green->blue, or blue->red transition as +$1/3, and each red->blue, blue->green, or green->blue transition as -$1/3. The result will be your total profit, no matter how the inner tiles are colored.
Thus, your profit is maximized by painting red->green->blue->red->green->blue-… clockwise along the outside border, netting you floor(perimeter/3) dollars.
There is a simple reason why things work out this way (nothing high-falutin’). In case anyone would enjoy figuring it out, I’ll leave that mystery open for now (as well as the implications for playing on a closed surface like a tetrahedron, cube, or soccer ball); if not, I’ll post the punchline this evening.
Oh, one other technical point which septimus already saw:
On some boards, there may be tiles which hang onto the border but never actually touch two other tiles (for example, the two red squares at the bottom corners here); these are, as septimus put it, “useless tits”, since they touch no points at which dollars may be gained or lost. Just ignore these when painting red->green->blue->… along the outside border.
I’ll try to make a harder one but … it’s my bedtime.