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Racing Cubes
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A Pip Is a Pip Is a Pip
Or Is It?
Kent Goulding, 1981
Las Vegas Backgammon Magazine, September 1981
I don't recall where I encountered the following position. It is one of several interesting riddles which pop up from time to time. This problem reminds me a lot of the famous "is five men off with one on the bar a take or pass?" hustle. It seems that nobody remembers just what happened when that famous backgammon master hustled the Mayfair Club in New York and walked out with a bundle. Was it
five checkers off or
six checkers off? Was it a drop, or a take? Ask any two players who were there and you'll get two different answers. So it is with this position.
The position is usually presented with Black (even numbered pips) on roll against White (odd numbered pips). Even is on roll, but Odd is fifteen pips ahead — or is he? The point is that whenever Black (even) rolls and odd number he can fill in one of the holes on the odd points. When White (odd) rolls an even number he retains his gaps and must "waste" that part of his roll. So, who is winning? Should Black double? Should White accept? Should White beaver? Should White pass?
Perhaps if we break the position into three separate problems it will help. Let's try — Divide the position into three parts as follows:
In Problem 1, White is ten pips ahead. It should be obvious that many of these ten pips are in fact wasted. That is if White rolls a 3, 4, 5, or 6 it has the exact same result. Any roll over 3 is "wasted." Black also has some waste, but not as much as White. As it turns out White will actually win just over fifty percent of the games with no cube involved. It would be an error for Black to double here. White should beaver.
Position 2 still shows White ten pips ahead. Much of White's waste has gone by the wayside, though. Now White can play any number that misses (1, 2, or 4) to an open point. Indeed, in this position White gets almost full value out of his ten pip lead. If the game is played to a conclusion (no cube) White wins almost fifty-nine percent of the games! Certainly Black would be foolish to double. White would have an easy beaver and probably redouble on the very next roll!
Position 3 again shows that White is ten pips ahead. How what is going on? White misses on 2's, 3's, or 4's; but most of these numbers fill in vacant points and are therefore not toally wasted. In fact, White still has most of his ten-pip lead and will win well over half of the games (57%+) with no cube. Again, if Black has access to the cube he would be foolish to double. White would be an even an even bigger than 57–43 favorite if he had the cube, and would beaver instantly.
Does any of this help you to understand the original problem? Does any of this have anything to do with the original problem? Is a pip a pip? How much does waste matter in a close race?
Certainly you should see that the original position has a lot of waste for White. But does it have enough to overcome the apparent fifteen-pip lead? In each of our three mini-problems Black would have been quite foolish to double. In each case White would be perfectly safe to beaver. What do you think now about the main problem?
At the recent tournament in Clearwater, Florida, this position reared its ugly head. A well-known, respected New Yorker, along with a room full of kibitzers, informed me that Black had a double and that White should pass! I was somewhat amazed, but volunteered to donate some of my money by simply taking White with the cube. (If, in fact, it was a clear pass, they should have been willing to give the cube on 2 plus one point per game.) They were delighted to hustle me with just giving up the cube.
What do you think? Was I foolish? Were they foolish? Your comments are welcome. I'll publish my findings in a later issue.
[The following answer appeared in the April 1982 issue of
Las Vegas Backgammon Magazine.]
The question asked is: with Black owning the cube and White on roll in the following position, which side is winning? I promised my answer in a later issue, so here it is.
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White to roll.
Which side is favored?
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Take the fifteen pip lead.
Even with no cube in play you will be a 55–45 favorite. Holding the cube would make it a complete slaughter. The fifteen pips far outweigh the "wastage" suffered when you roll even numbers. Try it a few times — you'll soon be convinced.