| Backgammon Variants |
The term “contact” is well known to backgammon players. When you lose contact with your opponent, it means all of your checkers have moved past all of your opponent’s checkers and the game has transitioned into a “pure race” with no more hitting or blocking.
But what does contact mean exactly? Let’s look at some possible definitions.
Overlapping Contact
A formula I use in computer programming to determine if a position has contact or not is the “overlap formula.” It works by taking the point number of black’s farthest-back checker and adding to it the point number of white’s farthest-back checker. If the total is greater than 25, then the position is deemed to have contact.
Here is an example:
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20 v 6: Black to roll |
Both players have one checker remaining, black on his 20 point and white on his 6 point. Adding 20 + 6 gives 26, and since 26 is greater than 25 the formula says this position has contact.
And the formula works, at least most of the time. But let’s look more closely.
No Possible Contact
Consider this position:
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1,1 v bar: Black to roll |
The overlap formula says this position has contact because 1 + 25 = 26 which is more than 25. But is there really contact here?
It’s black’s turn. Black can’t hit because white’s only checker is on the bar. And white can’t hit because black will have borne off and the game will be over before white even gets a chance to roll. There is no way for either side to hit or block the other, ever. There is “no possible contact” in this position.
No Practical Contact
How about this position?
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2,1 v bar: Black to roll |
At first this looks like the previous position. Black will have borne off his checkers before white even rolls, so there is no possible hitting or blocking here. Or is there?
Suppose black rolls 2-1 but, instead of taking two checkers off, he plays his 1 first, moving 2/1 and then bears off using his 2. That would leave a black blot on the ace point! We can’t really say the game is over yet.
But really? Nobody is ever going to play a 2-1 this way. For all practical purposes, this game is over. I call this “no practical contact.”
No Significant Contact
Here is an example of a position where there can still be hitting but the hitting has no effect on the outcome of the game.
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3 v 24: Black to roll |
If black rolls 2-1, it doesn’t matter whether he hits white’s checker on his way to bearing off or not. Black wins the game either way. The only difference is whether white’s checker ends the game on the bar or on p24.
So there still can be hitting in this game, but the hitting is of no consequence. Positions like this have “no significant contact” because whatever contact they have is has no affect on the outcome of the game.
Block-Only Contact
There is another interesting type of contact. There are no examples from hypergammon that have “block-only” contact, so here is an example from backgammon.
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bar v 1x4: Black to roll |
There is no hitting left in this game: White can’t hit because black is on the bar and black can’t hit because white always has an even number of checkers on his ace point.
What about blocking? Consider this scenario: Black rolls 1-1 and dances; white rolls and takes two checkers off; black rolls 6-6 and enters bar/1; white rolls and wins the game.
If black’s original 1-1 roll hadn’t been blocked, the game would have been different. If black had somehow been able to enter with his 1-1, then his subsequent 6-6 would have borne off his checker and won him the game. So, while there is no hitting left in this game, there is blocking. And the blocking plays a part in determining the outcome of the game.
Which Positions Have Contact?
Last time, I gave a ten positions and asked which have contact and which do not.
 a. 12,5,1 v 12,1,1 |  b. 1,1 v bar |
 c. 3 v bar |  d. 2,1 v bar |
 e. 3,1 v 23 |  f. 4 v 22 |
 g. 5,1 v 22 |  h. 3 v 24,1,1 |
 i. bar v 1,1 |  j. 13 v 13,13 |
Here are my answers according to the definitions of contact I gave earlier.
| Overlap | No Possible | No Practical | Block-Only | Significant | ||
| a. | 12,5,1 v 12,1,1 | N | N | N | N | N |
| b. | 1,1 v bar | Y | N | N | N | N |
| c. | 3 v bar | Y | N | N | N | N |
| d. | 2,1 v bar | Y | Y | N | N | N |
| e. | 3,1 v bar | Y | Y | N | N | N |
| f. | 4 v 22 | Y | Y | N | N | N |
| g. | 5,1 v 22 | Y | Y | N | N | N |
| h. | 3 v 24,1,1 | Y | Y | Y | N | N |
| i. | bar v 1,1 | Y | Y | Y | Y | N |
| j. | 13 v 13,13 | Y | Y | Y | Y | Y |
Measuring Contact
Can you measure how much contact a position has? It seems you should be able to. A game that is likely to see a lot of hitting and blocking could be considered to have a lot of contact, and a position that rarely sees any hitting or blocking could be said to have very little contact.
With that in mind, let me ask:
- What position in hypergammon has the most contact?
- What position in hypergammon has the least contact (excluding positions that have no contact at all)?
Any ideas?
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