This is the twelfth in a series of posts about the game of hypergammon. The previous posts are [Hyper 01], [Hyper 02], [Hyper 03], [Hyper 04], [Hyper 05], [Hyper 06], [Hyper 07], [Hyper 08], [Hyper 09], [Hyper 10], and [Hyper 11].
How long does a typical game of hypergammon last? That is the topic of today’s post.
We will only look at cubeless hypergammon for now, so we don’t have to worry about games ending with a “double/pass.” We only count games played to conclusion, i.e., where the winner has borne off all his checkers.
Shortest Possible Game
Each player in hypergammon starts with a pip count of 69. The most pips you can roll in one turn is 24, so you need at least three rolls (24 pips x 3 = 72 pips) to bring your checkers home and bear them off.
Can the winner of the opening roll get his checkers off in 3 rolls? No, because his first roll can’t be doublets. The most he can roll on his first roll is 11 pips, so it will take him a total of 4 rolls to bring his checkers home and bear them off.
What about the loser of the opening roll? Can he bear off in 3 rolls? Yes he can, if he rolls three double 6’s in a row. He just needs to hope that there are no blocks in his way.
The shortest possible game, then, is 6 rolls. Here is an example of how it can be done:
| Black | White | |||
| 1. | 6-5: | 23/17, 22/17 | 6-6: | 24/12, 23/11 |
| 2. | 6-6: | 17/5(2) | 6-6: | 22/10, 12/6, 11/5 |
| 3. | 6-6: | 24/off | 6-6: | 10/4, 6/off, 5/off, 4/off |
Longest Possible Game
It turns out there is a tie for the longest possible game of hypergammon. One of the shorter contenders is illustrated below. This game starts with the following sequence of rolls: 5-1, 6-6, 5-5, 6-6, 4-4, 2-2.
Played perfectly, these rolls produce the following game:
| Black | White | |||
| 1. | 5-1: | 24/19, 23/22 | 6-6: | 24/12, 23/17, 22/16 |
| 2. | 5-5: | 22/12(2) | 6-6: | 17/5, 16/4 |
| 3. | 4-4: | 12/4(2) | 2-2: | 12/6*, 6/4 |
That brings us to this position:
 | Black to roll |
At this point, the dice seem to get stuck and begin spewing an endless supply of double 5’s. I will leave it as an exercise to the reader to play the game out and see how it ends.
Average Length Game
The average length of a game of hypergammon is 24.5822837179629 rolls. (Or at least it is somewhere around that number.)
To get this approximate number, we assume (as usual) that both players are playing perfectly. (If the players were trying to lose instead, you’d get a very different answer!) But there is still some wiggle room here.
To get the precise number I gave above, we need one additional rule about how the game is played. To illustrate, look at this position:
 | Black rolls 6-1 |
Black has two ways to play his 6-1: He can bear off both checkers and win the game immediately, or he can play 3/2/off and make white suffer a little longer. Both plays have the same equity (black wins either way), but the second play makes for a game that is 2 rolls longer.
To choose between these plays, I use the following rule: “Among multiple plays with the same equity, choose the play that has a lowest effective pip count.” Applying this rule to the position above means that black should bear off both his checkers now.
Last time, I gave as a clue that “it takes an average of 13.45010 rolls to bring your checkers home and bear them off.”
This tells you that the average length of a game has to be less that 13.45 × 2 = 26.9 rolls. That’s because the winner of the game will be luckier than average, so it will take him less than 13.45 rolls to get his checkers off. And the loser never throws more rolls than the winner. So the average number of total rolls in a game has to be less that 26.9.
We saw that the actual average is about 24.58 rolls. The difference between 26.9 and 24.58 is 2.32 rolls. This is the average number of rolls it will take the loser to finish bearing off his checkers if he were to continue playing.
Median Length Game
The number of rolls that most evenly divides hypergammon game lengths into the shorter ones and the longer ones is 22 rolls.
- 44.94% of all games take less than 22 rolls to finish
- 49.73% of all games take more than 22 rolls to finish
Most Common Length Game
The following histogram show the relative frequency of the different lengths of hypergammon games.
| 6 rolls | 0.00000 | |
| 7 rolls | 0.00005 | |
| 8 rolls | 0.00010 | |
| 9 rolls | 0.00088 | |
| 10 rolls | 0.00142 | |
| 11 rolls | 0.00576 | |
| 12 rolls | 0.00768 | |
| 13 rolls | 0.01918 | |
| 14 rolls | 0.02302 | |
| 15 rolls | 0.04231 | |
| 16 rolls | 0.04461 | |
| 17 rolls | 0.06044 | |
| 18 rolls | 0.05774 | |
| 19 rolls | 0.06642 | |
| 20 rolls | 0.05928 | |
| 21 rolls | 0.06058 | |
| 22 rolls | 0.05329 | |
| 23 rolls | 0.05202 | |
| 24 rolls | 0.04530 | |
| 25 rolls | 0.04323 | |
| 26 rolls | 0.03742 | |
| 27 rolls | 0.03494 | |
| 28 rolls | 0.03036 | |
| 29 rolls | 0.02800 | |
| 30 rolls | 0.02447 | |
| 31 rolls | 0.02241 | |
| 32 rolls | 0.01962 | |
| 33 rolls | 0.01784 | |
| 34 rolls | 0.01558 | |
| 35 rolls | 0.01414 | |
| 36 rolls | 0.01241 | |
| 37 rolls | 0.01114 | |
| 38 rolls | 0.00982 | |
| 39 rolls | 0.00886 | |
| 40 rolls | 0.00780 | |
| 41 rolls | 0.00701 | |
| 42 rolls | 0.00615 | |
| 43 rolls | 0.00552 | |
| 44 rolls | 0.00487 | |
| 45 rolls | 0.00436 | |
| 46 rolls | 0.00384 | |
| 47 rolls | 0.00343 | |
| 48 rolls | 0.00301 | |
| 49 rolls | 0.00271 | |
| 50 rolls | 0.00236 | |
| 51 rolls | 0.00211 | |
| 52 rolls | 0.00187 | |
| 53 rolls | 0.00167 | |
| 54 rolls | 0.00146 | |
| 55 rolls | 0.00129 | |
| 56 rolls | 0.00116 | |
| 57 rolls | 0.00103 | |
| 58 rolls | 0.00091 | |
| 59 rolls | 0.00082 | |
| 60 rolls | 0.00072 | |
| 61 rolls | 0.00063 | |
| 62 rolls | 0.00056 | |
| 63 rolls | 0.00049 | |
| 64 rolls | 0.00044 | |
| 65 rolls | 0.00040 | |
| 66 rolls | 0.00036 | |
| 67 rolls | 0.00031 | |
| 68 rolls | 0.00026 | |
| 69 rolls | 0.00024 | |
| 70 rolls | 0.00021 | |
| 71 rolls | 0.00019 | |
| 72 rolls | 0.00017 | |
| 73 rolls | 0.00015 | |
| 74 rolls | 0.00013 | |
| 75 rolls | 0.00011 | |
| 76 rolls | 0.00011 | |
| 77 rolls | 0.00009 | |
| 78 rolls | 0.00008 | |
| 79 rolls | 0.00007 | |
| 80 rolls | 0.00007 | |
| 81 rolls | 0.00006 | |
| 82 rolls | 0.00005 | |
| 83 rolls | 0.00005 | |
| 84 rolls | 0.00004 | |
| 85 rolls | 0.00004 | |
| 86 rolls | 0.00003 | |
| 87 rolls | 0.00003 | |
| 88 rolls | 0.00003 | |
| 89 rolls | 0.00002 | |
| 90 rolls | 0.00002 | |
| 91 rolls | 0.00002 | |
| 92 rolls | 0.00001 | |
| 93 rolls | 0.00001 | |
| 94 rolls | 0.00001 | |
| 95 rolls | 0.00001 | |
| 96 rolls | 0.00001 | |
| 97 rolls | 0.00001 | |
| 98 rolls | 0.00001 | |
| 99 rolls | 0.00001 | |
| 100 rolls | 0.00001 | |
Six-roll games are possible, but they are so rare that they don’t even show up on the chart. Games longer than 100 rolls are also extremely rare.
The most common length of game is 19 rolls.
You might expect that the second-most-common length would be either 18 rolls or 20 rolls, but it is not. Second and third place go to “21 rolls” and “17 rolls” respectively.
The reason is that the winner of the opening roll wins the game more often than the loser of the opening roll. When the winner of the opening roll wins the game, there are an odd number of rolls because the same player throws both first and last.
The Five M’s
Statisticians refer to the average of a distribution as its “mean” and they call the most common value of a distribution the “mode.” Here are the five m’s of hypergammon game lengths:
| Minimum | 6 | |
| Mode | 19 | |
| Median | 22 | |
| Mean | 24 | .6 |
| Maximum | ∞ |