Hyper Fun 02. Counting Positions
Tom Keith, 2018

Returning to the Start

Last time we asked for a sequence of rolls and plays that allow a game of hypergammon to return to the starting position.

starting position
Black’s side of board    

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; they all involve variations on the following theme. Suppose black wins the opening roll.

  1. Black moves a checker from b24 or b22 to b18 or b13.
  2. White rolls 5-5 and hits two black blots using his checker on w22.
  3. Black enters his two checkers and, in so doing, hits the blot left by white on b23.
  4. White rolls 2-1 and reenters his checker back onto w22. The starting position is restored.

Here is one way to do it:

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.

Number of Different Hypergammon Positions

It has long been known that backgammon has 18,528,584,051,601,162,496 “legal” positions. (A “legal” position is one where each side has up to 15 checkers on the board, and no point is occupied by checkers of both colors.)

What about hypergammon? How many different positions does it have? An easy way to find out is by writing a computer program to do it. Here is an example written in Swift:

It turns out the answer 7,959,904.

You can also get this answer mathematically, but the method is a little tricky and tedious, so I will describe it in a separate post.

7,959,904 may sound like a lot. After all, you could never expect to memorize that many positions. But it is positively miniscule compared with the number of backgammon positions.

Why this is important is that it means hypergammon can be solved by computer — we can calculate the exact equity of every possible position. That’s not true of backgammon. For most backgammon positions, computers can only guess at the correct equity.

There is a slight wrinkle we should iron out here. 7,959,904 is the number of “legal” positions. The only problem is that some of these positions cannot be reached!

For example, there is no way in hypergammon for both players to end up with all of their checkers on the board at the same time.

This position cannot be reached

You can see why if you try to imagine what position and roll led to this situation. The sixth checker on the bar must have been hit by somebody; but how could that happen without the hitter entering one of their own checkers. This impossible-to-reach position is included in 7,959,904 figure above.

Another position that can’t happen is “all checkers off the board.”

This position can’t happen

The reason this can’t happen is that as soon as one player gets all his checkers off the game is over. But “all checkers off the board” is also included in the count above.

These are two examples of unreachable positions. Are there others? How many legal hypergammon positions can actually be reached from the starting position?

Take a guess. What percent of the 7,959,904 hypergammon positions are reachable? Choose among the following possibilities:

  1. 99.997%
  2. 98%
  3. 89%
  4. less than 50%

I will post the answer next time.

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Hyper Fun 03

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