Hyper Fun 12. Length of a Game
Tom Keith, 2018

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:

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 7 rolls 5e-05 8 rolls 0.0001 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 0.0453 25 rolls 0.04323 26 rolls 0.03742 27 rolls 0.03494 28 rolls 0.03036 29 rolls 0.028 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 0.0078 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 0.0004 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 9e-05 78 rolls 8e-05 79 rolls 7e-05 80 rolls 7e-05 81 rolls 6e-05 82 rolls 5e-05 83 rolls 5e-05 84 rolls 4e-05 85 rolls 4e-05 86 rolls 3e-05 87 rolls 3e-05 88 rolls 3e-05 89 rolls 2e-05 90 rolls 2e-05 91 rolls 2e-05 92 rolls 1e-05 93 rolls 1e-05 94 rolls 1e-05 95 rolls 1e-05 96 rolls 1e-05 97 rolls 1e-05 98 rolls 1e-05 99 rolls 1e-05 100 rolls 1e-05

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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