i couldn't find an efficient way to copy/paste the position here, so go take a look at position 40 in this annotated game:

http://www.bkgm.com/matches/woba-1.html

15 vs 14 men left?
crossover?

after the 21/9, ace-ace is the only roll that sucks for red and white getting hit would open up too many gammon chances?

A crossover means the crossing of a quadrant of the board. Basically, since a 6 will only let you cross exactly one quadrant, the number of crossovers is the minimum number of dice you need to play to bear all of your checkers off.

So 14 crossovers means that it takes at least 14 dice to bear 'em all off. That is normally (i.e. barring doubles) equivalent to 7 rolls. The guy with 15 crossovers to make needs for sure more than seven rolls to bear off, if he doesn't roll doubles along the way.

Now you can have situations where you lead the pip count but trail in crossovers and that's when it might become all tricky (-:

To answer your question about the article: If white leaves the checker in red's homeboard, white can afford to hit this checker with a roll like 61 or 51. This would mean that he could only bear one checker off. Now because he has an uneven number, 15, of crossovers, he would need 8 rolls no matter if he hits and bears only one checker off or doesn't hit and bears two checkers off. For this reason white should have run with both checkers, not leaving the shot. Leaving the shot only increases the chance of losing a gammon with no upside.

They're talking about white's crossovers - the issue here is that 12/9 doesn't really get white much compared to 21/18. Meanwhile white can't stop red anyway, so should run as efficiently as possible.

If red had an even number of checkers, leaving the blot on 21 might be right because it forces red to waste a die on rolls like 3-2, which costs him an extra turn bearing off. But because red is odd, he can afford to waste one.

No I believe Kit is talking explicitly about the white crossovers. You are right that the play of the last trey wastes a crossover. Kit recommends running with both checkers which implies that the last trey is to be played 21/18.

BUT:

So he is pretty much saying he is gratefully playing a roll like 61 pick and pass (5*4/off) because he would not lose a roll by hitting, compared to if he had only 14 crossovers to make instead of 15.

Isn't that what I said? You're the one talking about red's crossovers. (And I think it's kind of weird to think about crossovers after bearin is done anyway.)

White doesn't really care about this, though - in fact this is a further advantage to running. The only reason for white to sacrifice running efficiently in order to leave a blot, as Jeremy does, is to gain an extra roll when red has to move the checker on his five point to avoid leaving a shot. Which doesn't work here because red is odd.

Hmm, yes that's what you said. The problem is that I messed the colors up in my last post. Switch red and white. I was talking about the crossovers of the bearing-off player (Kit).

So, for this not to happen again, I'll include the position now... (made a screen shot).



Actually it is not weird to think about crossovers during a bear-off. The scenario where leaving the shot makes the BIG difference is this:

1. White (Bagai) plays 21/9 instead of 21/12 21/18.
2. Red (Woolsey) rolls a 1 (whopping 11 times)
3. Red hits and then bears the hitting checking off. This does not cost him a roll, because he had an uneven number of crossovers. 11 is played pick and pass 5*4/3, and then 2/off.
4. White dances like 2 or 3 times
5. Red turns on the turbo, rolling a double or so. He ends up winning the gammon.

Had White played 21/12 21/18 losing a gammon is still possible in case of cold dice, but it takes White only 5 crossovers to bear one checker off. As played, White loses serious crossovers in case he is hit. Hitting has no downside for Red because whether he hits the blot or not doesn't affect the overall number of rolls he needs to bear the checkers off. Red will win more gammons but doesn't really have to pay something in return.

Oh, yes. Obviously 21 and 31 don't hit and bearoff one checker without leaving a shot. But for all other rolls containing an ace (61,51,41,11 - still 7 rolls) the scenario I described is serious for white.

Coming back to this. I think we mean the same here, that's true. Red can afford it. The point Kit was making is that the play of leaving the shot could probably have made sense if hitting would indeed lose a roll. Then you re-enter with 65 and have speeded up.

In the situation as it was, Red would only have to slow down and waste a roll in order to hit if he rolls 31 or 21. Kit argues that this is for sure not worth it.