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.  65:  23/17, 22/17  66:  24/12, 23/11 
2.  66:  17/5(2)  66:  22/10, 12/6, 11/5 
3.  66:  24/off  66:  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: 51, 66, 55, 66, 44, 22.
Played perfectly, these rolls produce the following game:
Black  White  
1.  51:  24/19, 23/22  66:  24/12, 23/17, 22/16 
2.  55:  22/12(2)  66:  17/5, 16/4 
3.  44:  12/4(2)  22:  12/6*, 6/4 
And we arrive at this position:
 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 61 
Black has two ways to play his 61: 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 tiebreaking 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  
Sixroll 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 secondmostcommon 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  ∞ 

