Here are some new and updated numbers on Treehouse. These supercede my previous numbers, which included several miscounts. First, a brief explanation of terminology: - "moves" means the (number of) pairs of positions such that the type of move in question allows one to move from the first position to the second - "average out-degree" basically is how many positions you can move to (assuming positions to move from are equally likely) - "no-move positions" is the number of positions where you might be able to move the house (i.e. you can't change your own trio) - "unreachable" positions are positions you can't move to (with the particular type of move) That said, here are some numbers: positions: 204 TIP --- moves not shared with any: 84 moves shared with AIM: 348 moves, total: 432 average out-degree: 2.117647 no-move positions: 48 unreachable positions: 24 AIM --- moves shared with TIP: 348 moves not shared with any: 360 moves shared with DIG: 336 moves, total: 1044 average out-degree: 5.117647 no-move positions: 6 unreachable positions: 6 DIG --- moves shared with AIM: 336 moves not shared with any: 558 moves shared with HOP: 90 moves, total: 984 average out-degree: 4.823529 no-move positions: 24 unreachable positions: 48 HOP --- moves shared with DIG: 90 moves not shared with any: 594 moves shared with SWAP: 180 moves, total: 864 average out-degree: 4.235294 no-move positions: 48 unreachable positions: 48 SWAP ---- moves shared with HOP: 180 moves not shared with any: 432 moves, total: 612 average out-degree: 3.000000 no-move positions: 0 unreachable positions: 0 WILD moves: 2982 If we order them by total moves, we get this: AIM: 1044 DIG: 984 HOP: 864 SWAP: 612 TIP: 432 If we perturb the numbers a bit (by 36 at most, which is less than 6% of the relatively most perturbed number) and divide them by their greatest common divisor, we get the sequence [4, 6, 8, 9, 10]. That is, the most flexible* move (AIM) is about 2.5 times as flexible as the least flexible (TIP), and the more flexible types are less different in flexibility than the least flexible ones. (* if we measure flexibility as simply the number of moves of each type, i.e. taking each position as equally likely to be in, which probably isn't true) I have also confirmed what I have previously hypothesized and argued: that it takes four moves to go from the starting position to the ending position, but only two moves in the other direction. More completely, the numbers of pairs of positions such that it takes d moves to go from the first to the second, grouped and sorted by d, are these: 0: 204 1: 2970 2: 14706 3: 19716 4: 3996 5: 24 (for d=0, this obviously has to be the number of possible positions) Note that these numbers are divisible by 6, as there are 6 different size permutations; in particular, there are only 4 "truly different" pairs of positions which are 5 moves apart. They are: - 1L 2U 3U to 2R 3L 1R - 1L 2U 3R to 2R 3L 1R - 1L 2U 3R to 3L 1R 2L - 1U 2U 3R to 3L 1R 2L (by this notation, the default house is 2L 1U 3R; I hope this explains it) Even if one feels that games are too short, we see that the only way to make the shortest games longer means having to remember some not-so-nice positions. So kudos to Andy for making a good design choice :) That's it for now; If anyone wants the 318 lines of python code which does this counting, let me know. I think I need to move from looking at the game as a graph to looking at the game as a markov decision process to come up with more interesting numbers. I might, but I make no promises. If anyone wants to beat me to it, feel free :) -- Jonas Kölker <jonaskoelker@xxxxxxx> <URL:http://jonaskoelker.ignorelist.com>
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