| Backgammon Variants |
How long does a typical game of hypergammon last? That is the topic of today’s post.
For now, we will look at cubeless hypergammon, so there is no possibility games ending in “double/pass.” All games are played until one side or the other has borne off all their checkers.
Shortest Possible Game
Each player in hypergammon starts three checker that have to move a total of 69 pips. The most pips you can roll in one turn is 24, so you need at least three rolls (24 pips × 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 the first roll can’t be doublets. The most pips you can roll on the opening roll is 11, so the opening roll winner will need at least 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.
| 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 off 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 |
And we arrive at this position:
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| Black to roll |
At this point, the dice begin spewing out an endless supply of double 5’s. I leave it as an exercise to the reader to play out the rest of the game and see how it ends.
Average Length Game
Last time, I noted that “it takes an average of 13.45010 rolls to bring your checkers home and bear them off.” So the average length of a game must be less that 13.45 × 2 = 26.9 rolls 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.
It turns out the average length of a game of hypergammon is 24.5822837179629 rolls. To get this number, we assume (as usual) that both players are playing perfectly. (If the players were trying to lose instead, we would get a very different answer!)
The difference between 26.9 and 24.58 is 2.32 rolls. This is the average number of rolls it would take the loser to finish bearing off his checkers if he were to continue playing after the game was over.
To get the number 24.5822837179629, there is one additional assumption we had to make 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 a single game either way), but the second play yields a game that is 2 rolls longer.
To decide between two such plays, we use the same tie-breaking rule we used in Hyper Fun 10:
If two plays are tied in equity, choose the play that minimizes total epc.
Applying this rule to the position above means that we assume black plays 3/off, 1/off and bears off both of his checkers now.
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 this 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 of game 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 | ∞ |
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