The rules of this game can be found here. However, I tested a version in which redeals are not allowed. It turns out that Gaps is winnable surprisingly often (about 6 games out of 7), even without allowing redeals! In addition, a "fixed suits" version (also not allowing redeals) was tested, in which each Two can be moved to only one row which corresponds to a pre-determined suit.
| Fixed Suits? | Deals Played | Games Won | Win Rate % |
|---|---|---|---|
| No | 2,100 | 1,801 | 85.697 ± 1.495 |
| Yes | 250,000 | 62,127 | 24.851 ± 0.169 |
The winnability of every deal tested was successfully determined; every deal not won was a proven loss (no inconclusives).
Gaps can also be played with fewer than 13 ranks. I determined win rates (for both fixed suits and standard) using twelve or fewer ranks. For comparison purposes, the following table also includes the results above for 13 ranks.
| Ranks | Fixed Suits Win Rate % | Standard Win Rate % |
|---|---|---|
| 2 | 86.724 ± 0.021 | 100 |
| 3 | 59.088 ± 0.096 | 94.885 ± 0.043 |
| 4 | 44.362 ± 0.097 | 89.326 ± 0.061 |
| 5 | 35.653 ± 0.094 | 85.589 ± 0.218 |
| 6 | 30.327 ± 0.090 | 83.607 ± 0.725 |
| 7 | 27.110 ± 0.276 | 82.298 ± 0.748 |
| 8 | 25.216 ± 0.269 | 83.885 ± 1.609 |
| 9 | 23.763 ± 0.264 | 81.889 ± 1.685 |
| 10 | 23.558 ± 0.372 | 81.704 ± 1.651 |
| 11 | 23.520 ± 0.263 | 83.890 ± 1.570 |
| 12 | 24.260 ± 0.840 | 85.506 ± 1.503 |
| 13 | 24.851 ± 0.169 | 85.697 ± 1.495 |
It appears that the game becomes harder to win as you use more ranks, up through about ten, and then it becomes more winnable. This means that adding ranks makes the game harder to win in one way, but easier to win in another way, and the relative importance of these two consideraitons changes. Perhaps the explanation is that adding ranks increases the likelihood of having at least one blocking pattern present, but also potentially increases the opportunities for breaking free from these blocks.
I noticed that many of Solvitaire's solutions involve moving a Two from one row to another, even though it was already at the leftmost position of a row. So, I added options to my solver to see how many times it is necessary to move a leftmost Two from one column to another, and found that the percent of games that can be won without any such moves is higher than I expected: 62.4%! This means that for games that are winnable, there is a solution possible without these moves 3/4 of the time. But even when using just 7 or 8 ranks, the solver has found deals in which you have to move leftmost Two's 5 times!
For more information about this game and some of its variants, see Michael Keller's article Montana (Gaps) Solitaire.
For information about how I wrote my Gaps solver, see the technical details.
Any comments are welcome, please send to Mark Masten at:
Last modified February 9, 2024
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