One of my favorite backgammon books is “Backgammon Funfair,” by Ray Kershaw. Ray’s book is a collection of novelties and diversions related to backgammon. It is a fun read for anyone who likes mathematical curiosities or logic puzzles, especially if you already like backgammon.
Some of the interesting questions explored in Backgammon Funfair:
- What is the shortest possible (cubeless) game of backgammon?
- Are there any backgammon positions with zero equity?
- What position and roll yield the most number of plays?
- What position and roll are most affected by the Jacoby rule?
- What is the largest joker in backgammon?
- When are 3 checkers better than 2 in a bearoff race?
- Doubling paradoxes: Jacoby paradox, Kauder paradox, etc.
It’s a nicely done book, and there was obviously a lot of work put into creating it. If you enjoy these types of puzzles, I definitely recommend getting the book.
Some of the questions posed in Backgammon Funfair don’t have a conclusive answer. That’s because the game of backgammon hasn’t been solved yet. But there is a variant of backgammon, called “hypergammon,” which has been solved.
Hypergammon was devised in the 1980’s. It is played exactly as regular backgammon except that the players start with just 3 checkers each. The initial setup looks like this:
You might think that with just six checkers on the board this game would be pretty simple to play. But it is not as simple you might imagine. Hypergammon is a fast-paced game with many tricky checker and cube decisions along the way.
Over the next few weeks, I plan to post a series of articles on some of the strange properties of hypergammon and some interesting puzzles. Here is the first one:
Returning to the Start
The first chapter of Backgammon Funfair asks: “What are the fewest moves needed to return to the backgammon starting position?”
It’s a fun puzzle. You might be able to figure it out for yourself. It turns out it is also a fun question to ask about hypergammon.
Set up the board to the hypergammon starting position. Can you find a sequence of rolls that allow the two players, taking turns, to play in such a way that the game returns to the start? (Remember that the first roll cannot be doublets.)
Amazingly, it can be done in just 4 rolls. How do you do it? I’ll post the answer in a few days.