Wednesday, July 16, 2008
Wednesday Math, Vol. 32: The doubling cube
Since this is the 32nd Wednesday Math post and 32 is a power of 2, 2 to the fifth power to be exact, I'm going to write about the doubling cube, an addition to the rules of backgammon first introduced in the 1920's in New York City which helped revitalize the game and made it much more challenging. Backgammon is very ancient. It's hard to find good information on when people started playing by the modern rules we use now, but there are examples of variations of the game played in many cultures around the Mediterranean and east to Persia, and some of these date back to millenia before Christ was born.
The doubling cube is a way to increase the gambling action. Every game is played for a set stake, let's say a dollar. By most rules, if you win a gammon (this means winning by a lot), the game is worth two dollars and winning a backgammon (which means winning by a whole lot), the game is worth three dollars. So, without the doubling cube, the best or worst you could do in a single game is win or lose three times the set stake in a single game. Most people who play will play many games in a single sitting.
The doubling cube sits off to the side when the game begins. The faces of the cube have the numbers 2, 4, 8, 16, 32 and 64 on them, the first six positive powers of two. Let's say that I think I have an advantage as the game progresses. After my opponent finishes a move, but before I roll the dice for my move, I can offer the doubling cube to my opponent with the side that has the 2 on it face up. What this means is that I want to play for double the stakes. My opponent now has a choice.
Option #1: Refuse to play for more and resign the game.
Option #2: Accept the offer to play for more and take possession of the cube.
If I was correct and I have an advantage, why would someone take the cube and probably lose two dollars? Well here's the math.
If I truly have an advantage, I should win the game from that position more than 50% of the time. Let's say I should win 60% and my opponent should win 40%. In this case, the opponent should take the cube because it is a better play in the long run. Lemme 'splain.
Option #1: My opponent loses one dollar.
Option #2: 60% of the time, my opponent loses two dollars and 40% wins of the time wins two dollars. 40% - 60% = -20%. -20% x two dollars = -40 cents.
In the long run, option #2 loses 40 cents, while option #1 loses a dollar. Take the cube and the smaller long run loss. The cut-off point is if a player thinks the chance of winning is 25% or less. Then the cube should be declined.
The math gets a little trickier if we factor in gammons and backgammons, and if the game changes and my opponent gains the advantage, ownership of the cube means he or she has the right to double the stakes once again, so we would be playing for four times the original stake if I agree to the new situation. There's also the immediate re-double called a beaver, and the re-re-double called a raccoon, and an entire menagerie of more exotic and ultimately expensive options.
It's a really interesting concept and can be added to many gambling games, if that's what you are into. I make no moral judgments on this matter.
Yay, Flags of Many Lands™! Super yay, Albania! Now with a visitor from Albania, I have the whole map of Europe except for the little postage stamp principalities like San Marino, Andorra, Vatican City and Liechtenstein.
What brings an Albanian around to visit my happy little blog? Discussion of secret prisons! Cheerful little bastids, ain't they?