*Take It Easy* is a multiplayer solitaire game, and good for starting out
an evening of gaming when people are still filtering in and no one wants
to start a long game yet. Here's its Board
Game Geek page.

I believe that 307 is the highest possible score. Try to construct it! Want a hint, or just want the answer?

Because I am not good at this sort of puzzle, I solved it using a program
that creates many random boards, then breeds them using a simple
genetic algorithm.
Thus I cannot be sure that 307 is *really* the highest score, but
my program finds it in a few minutes, then fails to find any higher score.
And this guy
also claims it is.

If you play at random, the most common score is zero, which you get 50.95% of the time. Scores of 1 and 2 are impossible. Maybe surprisingly, the next most common score is 24, with runners up 12 and 27. It is quite unlikely to score 7 by accident. The mean score is 11.19. For scores above 60 or so, the probability of a given score decreases slightly worse than exponentially:

9.70│⠁ │ 8.94┤ 8.64┤⢀⢀⢀⠠⢀⢀⠠⠐⠠ 8.34┤ ⢀ 8.03┤ ⡀⡀ ⢀⠠⡠ ⡀ 7.73┤ ⠁⠈ ⠂⠁⠄⠄ l 7.43┤ ⡀⢀ ⡀⠐ ⡀ ⡀ o 7.12┤ ⠄ ⠄⡂⠂⠂⠐⠐⠄⠑⠠⠁⣐⢀⢁⠁⡀ g 6.82┤ ⠐ ⢀⡀⡀ ⠐ ⠐ ⢀ ⠄⡀⡀ 1 6.52┤ ⠈⠐⠉⠊⠠ ⠄ 0 6.21┤ ⠄ ⠁⠢⠔⢤ 5.91┤ ⠂ ⠜⣠ t 5.61┤ ⠑⢄ r 5.30┤ ⠋⠦ i 5.00┤ ⠱⢢ a 4.70┤ ⠱⢄ l 4.39┤ ⠣⡄ s 4.09┤ ⠈⠣⡀ 3.79┤ ⠉⢆ 3.48┤ ⠑⢄ 3.18┤ ⠲⡀ 2.88┤ ⠉⢆⡀ 2.57┤ ⠑⠄ 2.27┤ ⠈⢥⡀ 1.97┤ ⠰⣀ 1.66┤ ⠐⠤ 1.36┤ ⠈⠓⠄ 1.06┤ ⠐⢂⠂ 0.75┤ ⠁⠒⢁⢀ 0.45┤ ⢁⢀⡀⠁⡈ │ ⡀⢀⢀ 0┼──────────────────────────────────────────────────────────────╱ 0 189

The probability of scoring 100 or higher is 1 in 8400, and of scoring 150 or higher is 1 in 7 million. The highest score my program generated in 10 billion trials was 189, which it managed 3 times.

There is a sequel game called *Take it to the
Limit* which uses the numbers 1–12, giving 64 tiles instead of
27 in *Take it Easy*. It has two boards. One is a straight-forward
extension of *Take it Easy*: a hexagon with diameter 7. The other
uses three linked 3-hexagons. Some additional complications
are also added.

For the purposes of this page, let's ignore the complications
and focus on simple hexagonal boards of various sizes. Now, the
number of tiles on a board of radius *r* is 1 − 3*r*
+ 3*r*^{2}. ("Radius" means the number of tiles
from the center to the edge along a row of tiles, counting
the center tile.) The smallest set of tiles needed to play on
a board is the smallest cubic number equal to or greater than
this. Thus:

Radius | Board tiles | Total tiles | Remark |
---|---|---|---|

0 | 0 | 0 | The only losing move is not to play |

1 | 1 | 1 | Score is always 6… |

2 | 7 | 8 | Still a real game! (We playtested it.) |

3 | 19 | 27 | Standard Take it Easy |

4 | 37 | 64 | Take it to the Limit |

5 | 61 | 64 | |

6 | 91 | 125 | |

7 | 127 | 216 | |

8 | 169 | 216 | |

etc. | |||

As usual, the amount of fun goes to zero as the adjustable
parameter
goes either to zero or infinity. Based on Board Game Geek
rankings, it has a plateau at 3–4 (i.e. *Take it Easy* and
*Take it to the Limit* have about the same ranking).

I suspect that no radius over 4 is
playable, given that extra bonuses were deemed necessary to
make 4 a marketable game. Certainly I would think that
a radius 5 game (61 spaces on the board) would be very frustrating
if the minimum 64 tiles were used, whereas if 125 tiles were used,
*maybe* it wouldn't be totally awful. But it certainly
would be a long game. I suspect that one would have to pick a
few safe rows and focus on them or else risk a high probability
of scoring zero. If so, it's a bad game.

Repeat all the above exercises for *Take it Easy* for at least board
sizes 2 and 4, and maybe 5 and maybe make some general statements about size
N.