Last time I asked, what is the closest you can be to winning a game of hypergammon without actually winning it?
Bob Koca figured it out. Your highest probability of winning without having secured a victory is 0.99999995222306, and it can happen two ways:
||Black to roll|
||Black to roll|
In both positions, you (black) have 10 pips to go. If you get off in three rolls, you win the game. And that’s surely going to happen . . . unless you roll 2-1 three times in a row. The chance of that happening is 1/18 × 1/18 × 1/18 = 1/5832.
Even if you roll those three consecutive 2-1’s, white still has to get off in three rolls himself. The only way he can do that is by rolling 5-5 or 6-6 three times in a row (1/18 × 1/18 × 1/18), or by rolling the right combination of 6-6, 5-5, and 4-4 (5 ways × 1/36 × 1/36 × 1/36). The chance of either of those happening is 1/5832 + 5/46656.
Both must happen (you failing to get off in 3 rolls and white needing more than 3 rolls) for you to lose the game. The combined chances are:
That’s your opponent’s chances of turning the game around. Now there is a good hard-luck story!
“Wastage” in backgammon refers to pips you roll that go unused.
Your regular pip count (PC) is the number of spaces you must move your checkers to bring them home and bear them off. For example, at the start of a hypergammon game, your pip count is 24 + 23 + 22 = 69 pips.
Black’s pip count is|
24 + 23 + 22 = 69 pips
Your effective pip count (EPC) is the number of pips you expect to have to roll to bring all your checkers home and bear them off. Your EPC is 49/6 times the number of rolls you expect to have to throw (because 49/6 is the average number of pips per roll, about 8.167).
The difference, EPC − PC, is called wastage. Here is an example:
Black’s pip count in this position is PC = 3 (2 for the checker on the two point + 1 for the checker on the one point). Black has to move a total of 3 pips to get his checkers off.
Black’s effective pip count is EPC = 8.167. That’s because black always gets off in 1 roll, and 1 roll × 49/6 = 8.167. In other words, black expects to roll an average of 8.167 pips to get his checkers off.
The difference is black’s wastage:
Black will waste an average of 5.167 pips bearing his checkers off.
It is fun to look for positions that have very high or very low wastage. For example, here is the backgammon position that has the highest wastage:
||15 checkers on the ace point|
You can see why this position would be wasteful — every roll you throw, except for 1-1, wastes pips. Some rolls, such as 6-6, are hugely wasteful. Rolling 6-6 takes four checkers off, moving a total of four pips. But you normally expect 6-6 to move 24 pips, so 20 pips are wasted with this roll.
What does a low-wastage position look like? Here is the lowest wastage position in backgammon when all 15 checkers are still on the board:
||Trice triangle: 7-5-3|
In this position, higher numbers, such as 6, 5, and 4, bear off exactly, meaning no wastage. Smaller numbers can be used to fill the empty spaces on p1, p2, and p3, where they can be used later if you roll small numbers again. It is only when you get down to the last few checkers that you will actually have to waste pips.
Wastage in Hypergammon
What about hypergammon? What positions have the highest and lowest wastage? Let’s break the question down into three parts.
- One checker: What position has the highest wastage with 1 checker? What position has the lowest wastage?
- Two checkers: What position has the highest wastage with 2 checkers? What position has the lowest wastage?
- Three checkers: What position has the highest wastage with 3 checkers? What position has the lowest wastage?
Some of these are easy. Others take a little more thought. How many of these six positions can you figure out?