This is the second in a series of posts that I am writing about the game of hypergammon. (The first post in the series is here: [Hyper 01].)
Returning to the Start
Last time I asked if there is a sequence of rolls that allow you to return to the starting position.
 Black’s side of board | Hypergammon starting position |
It is not obvious that this can be done at all. But it can be, and it takes only 4 rolls. There are six different ways of doing it; Mike Clapsadle found one of them. They all involve variations on the following theme.
Suppose black wins the opening roll.
- Black moves a checker from b24 or b22 to b18 or b13.
- White rolls 5-5 and hits two black blots using his checker on w22.
- Black enters his two checkers and, in so doing, hits the blot left by white on b23.
- White rolls 2-1 and reenters his checker back onto w22. The starting position is restored.
Here is an example of how it can be done:
| | Black rolls 6-5. |
 | Black plays 24/13. White rolls 5-5. |
 | White plays 22/12*/2*. Black rolls 2-1. |
 | Black enters bar/24, bar/23*. White rolls 2-1. |
| | White enters bar/22, and the starting position is restored. |
How Many Different Positions Are There?
It has long been known that backgammon has exactly 18,528,584,051,601,162,496 legal positions. (By a “legal” position, I mean where each side has up to 15 checkers on the board, and no point is occupied by checkers of both colors.)
So how many different positions does hypergammon have? As a programmer, the easiest way for me to find the answer is by writing some computer code to calculate it. The following code is written in Swift:
It quickly calculates the answer to be 7,959,904.
There is also a mathematical way to get this number, but it is long and (for me) error-prone. I will describe it in a separate post.
7,959,904 may sound like a lot. After all, you could never expect to memorize so many positions. But it is a positively minuscule number compared with the number of different backgammon positions.
Legal versus Reachable
So that’s it. The number of positions in hypergammon is 7,959,904. There is only one problem with this number: It includes positions that are impossible to reach!
For example, it is impossible for both players to have all their checkers borne-off at the same time. (As soon as one player gets his checkers off, the game is over.) But “all checkers off for both players” is included in the count above.
Another unreachable example is 6 checkers on the bar. It is impossible, through normal play, for both players to have all their checkers on the bar at the same time. (To see why, think about what position might have preceded the last checker being sent to the bar.) But “six checkers on the bar” is included in the count above.
So how many of the 7,959,904 legal position can actually be reached from the starting position? Can you guess? Choose from the following possibilities:
- 99.7%
- 98%
- 89%
- less than 50%
I will post the answer next time.