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Posted By: Timothy Chow
Date: Thursday, 12 June 2014, at 12:55 a.m.

Nack recently composed a nice retrograde backgammon puzzle. More precisely, he composed what I would call a "proof game." In the chess problem literature, a proof game is a chess position accompanied by a stipulation such as, "This is the position after White's 10th move. How did the game go?" or more briefly, "PG 9.5". Some nice examples of (chess) proof games may be found here.

Proof games have turned out to be a very rich genre of chess problems. I'm curious as to how rich backgammon proof games can be.

If we follow the lead of the chess problemists, a crucial condition for a sound proof game is that the solution must be unique. Nack's backgammon problem doesn't meet this criterion, even with the somewhat inelegant side condition in "part deux", because some of the moves can be transposed, and that is a no-no in chess proof games. I'm sure that it's possible to come up with nontrivial examples of backgammon proof games with strictly unique solutions, but the question is whether the number of possibilities is too limited to be interesting.

To qualify as interesting, a chess proof game should have some thematic content. For instance, perhaps a superficial glance at the position suggests that neither of White's knights has moved, but in fact they must have exchanged places. Or perhaps one of the knights was captured and replaced by a promoted pawn (the Phoenix theme). Or maybe the knight took a long tour around the board before returning to its original square (a Rundlauf). These themes are aesthetically pleasing, because naively, you might think that you could "save time" by avoiding such a roundabout maneuver, yet the fact of the matter is that you can't.

Nack's puzzle does have some nice thematic content. Notably, it is necessary to vacate the 6pt and then reoccupy it. To achieve this effect, though, he had to introduce an inelegant side condition.

So perhaps a first challenge in the field of backgammon proof games could be:

Devise a backgammon proof game with a strictly unique solution (and no side conditions) that has a vacation/reoccupation motif.

Anyone up to the challenge? Bob Koca and Nack are the obvious contenders but everyone should feel free to have a try since I believe that this is a brand-new area. If vacation/reoccupation doesn't seem that interesting, then there are several alternative themes that come to mind. For example, maybe a particular checker doesn't seem to have moved at all, but it actually had to have moved, gotten hit, and then returned to its original location (i.e., a Rundlauf).

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