This is the third in a series of posts that I am writing about the game of hypergammon. (The first post is [Hyper 01] and the second is [Hyper 02].)
Unreachable Positions in Backgammon
A legal position in backgammon is a position that has up to 15 checkers for each side on the board with no point having two checkers of different colors. There are 18,528,584,051,601,162,496 legal positions in hypergammon.
Not all legal positions are reachable from the starting position, though, and here is an example:
 | Stalemate position |
You will never see this position in a normal game of backgammon (unless somebody makes an illegal move). It is an example of a legal position that can’t be reached.
It is an interesting question how many unreachable positions there are in backgammon. Nobody really knows and I don’t think I’ve even seen an estimate of this number.
Unreachable Positions in Hypergammon
What about hypergammon? How many unreachable positions does it have? I wrote a computer program to find out. It works like this:
From the starting position, the program looks at the 15 possible opening rolls, tries every way of playing each roll, and marks in a table which positions it can reach. There are a total of 86 different positions that can be reached in one roll from the starting position.
From those 86 positions, the program looks ahead one more roll and marks in the table the new positions it finds. This adds 12,343 positions to the total.
The program continues like this, each time looking ahead one more roll and marking the new positions. Here is the output:
The third roll of the game adds 196,150 new positions and the fourth roll adds another 1,814,304. After that, the number of new positions diminishes with each additional roll. And after 14 rolls, no new positions are added at all. The final total is 7,079,536 positions that can be reached.
In [Hyper 02], I asked what percent of legal hypergammon positions are reachable from the starting position and gave the following answers as possibilities: (a) 99.7%, (b) 98%, (c) 89%, and (d) less than 50%.
If you answered (c) 89%, then you are correct.
| Reachable positions | 7,079,536 | (88.94%) |
| Unreachable positions | 880,368 | (11.06%) |
| Legal positions | 7,959,904 | (100.00%) |
This surprised me a bit. I thought just about every legal position would be reachable somehow, that only a few weird cases would avoid having some path to them. But 11% unreachable positions is fairly substantial.
Does this 11% number have any implications for backgammon? Maybe backgammon has a similar ratio of unreachable positions. Or maybe there are many more or many fewer unreachable positions in backgammon. I don’t know.
A Hard-to-Reach Position
One thing that caught my eye from the output of the computer program above was that there are only four positions that take 14 rolls to reach. One of them is this one:
 | Hard-to-reach position |
Each side has two checkers on their own 8-point.
You can see why this would be a hard position to get to. Notice that both sides have one checker borne off already. That means at some point each side had all their checkers home (though obviously not at the same time), yet somehow their two remaining checkers are now back outside their home board. Not your everyday game!
Can you figure out how to get to this strange position? Can you do it in 14 rolls?