Gengulphus wrote:... By the way, I prepared a version of a differently rotated/reflected/colour-permuted version of the solution with coloured text to make the individual lines stand out better some time ago and have it on file - but it can't be spoiler-concealed because spoiler-concealment doesn't work on coloured text, due to it also using text-colouring. So it isn't appropriate here - but I might well post it later on in this thread.
Here it is:
/ R \
/ B \AAARMMM/ B \
/ G \AAABMMM/ R \FFFBMMM/ G \
\AAAGCCC/ B \FFFRMMM/ B \PPPGMMM/
/ G \FFFBCCC/ R \PPPBMMM/ G \
\FFFGYYY/ B \PPPRCCC/ B \AAAGMMM/
/ G \PPPBYYY/ R \AAABCCC/ G \
\PPPGYYY/ B \AAARYYY/ B \FFFGCCC/
\AAABYYY/ R \FFFBYYY/
And a follow-up puzzle:
The reason I tackled this problem (and discovered more-or-less inadvertently that it could be posed as a puzzle that had a pretty neat solution) was that I have a 'competitive solitaire' board game called "Take It Easy!". In it, each player has a small board with the blank hexagon-of-hexes on it and a set of 27 hex tiles, each with a set of coloured lines crossing it in the vertical, NW-to-SE and NE-to-SW directions. There are three colours for the vertical lines, three more for the NW-to-SE lines, and yet three more for the NE-to-SW lines, and there is one tile for each of the 27 combinations of one vertical colour, one NW-to-SE colour and one NE-to-SW colour. The game is played by having one person shuffle their set of tiles and place them in a face-down pile, while everybody else arranges their own sets of tiles so that they can pick out specific tiles reasonably quickly and easily (dividing them into three groups of 9 according to e.g. the vertical colour is generally good enough to let one pick out a particular colour). Then the person with the shuffled tiles takes the top tile from the pile, turns it over and calls out which tile it is. Everybody else takes the same tile from their own set, and everybody places their tile where they think it will best go on their board. When everyone has done that, the process is repeated until 19 tiles have been drawn and everybody's board is completed - note that once the next tile has been drawn, a tile may not be moved again.
The added element comes in the final scoring. Each colour is associated with a number in the range 1-9, with the numbers for the vertical tiles being 1, 5 and 9, those for the NW-to-SE colours being 3, 4 and 8, and those for the NE-to-SW colours being 2, 6 and 7. These are printed on the lines on the tiles, which is incidentally how one can tell which direction is vertical. Each line consisting entirely of a single colour scores the length of the line (i.e. 3, 4 or 5) times the colour's number, while each line with two or more colours on it scores nothing. Each player's score is their total for the fifteen lines, and the player with the highest score wins. As an example of the scoring:
/ 1 \
/ 5 \6661333/ 5 \
/ 5 \6665333/ 1 \7775333/ 9 \
\2225444/ 5 \7771333/ 5 \2229333/
/ 9 \7775444/ 1 \2225333/ 1 \
\6669888/ 5 \2221333/ 5 \6661444/
/ 9 \2225888/ 1 \6665444/ 9 \
\2229888/ 5 \6661888/ 5 \7779444/
\6665888/ 1 \7775888/
scores 3*8 = 24, 4*8 = 32, 0, 0 and 3*3 = 9 for the NW-to-SE lines, 0, 4*5 = 20, 5*1 = 5, 4*5 = 20 and 0 for the vertical lines, and 0, 0, 5*2 = 10, 4*6 = 24 and 3*7 = 21 for the NE-to-SW lines, making a total score of 165.
The follow-up puzzle has two parts:
* What is the maximum possible score?
* Suppose every number were one less, so that the numbers were 0-8, the vertical colours 0, 4 and 8, the NW-to-SE colours 2, 3 and 7, and the NE-to-SW colours 1, 5 and 6. What would the maximum possible score be in that case?
By the way, I mean the maximum possible scores in theory: to actually get either of them in practice would require a huge amount of luck, firstly to get one of the few combinations of 19 of the 27 tiles that one needs to (which is a tall order given that there are 27 CHOOSE 19 = 27!/(19!*8!) = 2220075 such combinations - i.e. in the same league as the chances of winning a Lottery jackpot with a reasonably small number of entries), and secondly because even if you do get one of those combinations, you've got to place most of the tiles with very incomplete information about what combination you're going to end up with...
As usual, answers with explanations of why they're the answers get more kudos than ones that are just the results of computer searches. But I will admit up front that I don't actually yet have fully-explained answers myself...