Racing Recube Study by O'Hagan

Lesson of the Week—Recubes and Racing

By John O’Hagan

In the position below, it is 4away/4away, Red holds a 2-cube and is on roll.  What is the proper cube action?


One should always start out knowing your opponent’s take point, and at this score, it’s 33% cubeless.  It’s also helpful to know that when the score is tied, your doubling window opens at as low as 50% and closes at the take point—so 50-67 percent is the window.

Next, it’s best to apply Woolsey’s Law—determine whether the taker has a take or drop, and then decide on the double or redouble.  Does White have a take here?  Red leads in the pip count by 68-70, White has 1 extra crossover, and both sides figure to waste about the same number of pips in bearing their checkers off.  This is clearly no redouble in a money game but what about at this match score?  What kind of winning chances does Red have here?

We can get a decent estimate of Red's winning chances in this position by comparing it to two benchmarks.  The first benchmark is that we know Red would be even money to win if he was on roll trailing 68-64 in the pip count.  The second one is that we know (per any of the standard racing formulas) that White would have a marginal take/pass decision for money if Red were on roll and ahead in the pip count by 68-77.  Red's cubeless chances would be around 78% in this latter case.   So if White's pip count was 64, he'd have 50% winning chances versus around 22% if he had 77.  He actually has 70 in this position so his cubeless chances should be around 6/13 of the way from 50% to 22% which is around 37%. 

How accurate is this 37% figure?  I see a couple of possible flaws.  The above benchmark comparison assumes a linear progression in White's winning chances as his pip count increases and this is not necessarily so.  Changing White's pip count from 70 to 71, for example, might diminish his chances more than if you increase it from 74-75.  So while the progression might not be 100% linear, it's probably real close and the error from assuming it's linear is small enough not to worry about IMO.

The other possible flaw is that the formula doesn't take White's extra crossover into account.  I don't know how much to deduct from White's chances for this extra crossover but over the board I would guess it's 1 or 2 percent so I would put White's chances at 35-36%.

Another method one could use to solve this problem is Kleinman's racing formula. Using his formula, the difference (D) is 6 and the sum (S) is 134.  D squared/S =around .27 which translates to the leader having approximately 64.5% winning chances (see 'the racer's edge' in his book Double-Sixes From the Bar for more details).  Adding another percent or two for the extra crossover gets you to around 65.5-66.5%.

So either method will lead you to the same conclusion:  White has a close take now but he might not if you gain a little on this next exchange.  The chance of Red gaining 2 or more pips on the next exchange is over 35% and White will no longer have a take if that happens, so now's the time to redouble.  Put another way, when you are this close to the take point, it is likely that you have too many market losers not to double.  The appropriate cube action is redouble/take.

Below is the extremeGammon evaluation which correlates to the above analysis.  Note that it is important to have the methodology to get these decisions right.  This position is a bare take.  If one checker were moved just a few pips either way it could easily be a big drop or a big no double, and of course, a different score and different take point would also affect the decision.