Once upon a time, there lived a Beaver in the Animal Kingdom.

The Beaver had just beat the highest difficulty level of Spider Solitaire – four suits sans undo. He felt he had played well after a difficult start, but it was hard to judge his overall ability at the game. After all, one wins and zero losses does not a large sample size make. And the fact none of his friends displayed any aptitude for the Royal Game certainly didn’t help. So, the Beaver decided to have a chat with his best friend, the Raccoon, who was known for his extensive knowledge of all things mathematics.

“It’s hard to judge your playing strength after one game,” said the Raccoon. “You need to play a large number of games to prove your victory wasn’t just beginner’s luck.”

“Suppose I played 129 games in a row,” replied the Beaver, plucking a three-digit number at random. “Then we can tally up my wins and losses and then we have a much better understanding of where I’m at.”

“Agreed,” replied the Raccoon. “Right now, the only thing we can agree on is you can play a hell of a lot better than I can.”

The Beaver chuckles, and he soon notices Captain Obvious is eager to join in the conversation.

“The only problem is it will take a long time to churn through 129 games,” says Captain Obvious. “Spider GM probably doesn’t wanna hear this but we all have better things to do in our lives than playing the Royal Game all day.”

“True,” says Raccoon. “Very True.”

*Hang on*, thinks the Raccoon. *129 happens to be a power of two plus one. This has me thinking – what if we can involve powers of two somehow? Let us say some games can be worth more than others. Suppose that each individual game was worth N victory points, where N was a power of two. A series of 129 games is equivalent to “First to 65 wins”. This should speed things up considerably. But Captain Obvious will gleefully point out Spider Solitaire is a game for one player, not two. Hang on (***thinks for a while***) I think I might have something.*

“Okay I have an idea,” says Raccoon.

“What is it?” asks the Beaver and Captain Obvious simultaneously.

“Let us pretend Beaver is the protagonist,” says Raccoon. “Only Beaver can move any cards. I am the Antagonist and I am willing Beaver to lose.”

Using a stick, the Raccoon sketches a hypothetical cube with all powers of 2 between 1 and 32.

“Initially, each game is worth 1 Victory Point. If Beaver thinks he has a good position, then he can double the stakes. I must concede 1 VP or agree to play on for 2 VP. Similarly, if I think Beaver has a poor position then I can double the stakes and Beaver has the same choice of refusing or accepting.”

“Sounds interesting,” says Beaver. “But if my game state were really bad, can’t you just double the stakes after every move? That wouldn’t be very interesting”

“That is correct,” replies the Raccoon. “Therefore, I propose another rule: if either side doubles the stakes and the opponent accepts then the opponent has the exclusive right to make the next double.”

“So that means, if I get a poor position, you double, I accept, then I turn the game around, then I can redouble and play for four VP?”

“Quite correct,” replies the Raccoon.

“Wait a minute,” says Captain Obvious. “If first to 65 wins then is it possible to get more than 65 if the doubling cube is more than 1?”

“Yes,” replies the Raccoon. “It doesn’t matter if you’re above 65 or exactly equal to 65. And before you ask, it’s perfectly legit for someone to double near the end of the match regardless of the game state because the math says he has nothing to lose.”

“Just to touch base,” says the rot13(fzneg nff) as he gleefully pokes the rot13(nff) of Captain Obvious, “does that mean only Beaver can moves cards, but both Beaver and Raccoon participate in cube-decisions.”

“That’s correct,” says Raccoon. “Even though I don’t move any cards, I can still participate in evaluating the winning chances of a given game-state. Win-win for everybody since I get a chance to improve my game as well.

This idea proved quite successful, and soon Raccoon was discussing the implications of the doubling cube with his friends, many of whom were also avid mathematicians. They had independently discovered some interesting theory and concepts such as market losers, the Crawford Rule, Jacoby Paradox, Woolsey’s Law for Doubling and so on. Not surprisingly, much of this theory is well-known to expert Backgammon players today.

For the record, the Beaver managed to win 66-42, although that may have been a function of Raccoon’s limited understanding of the Royal Game (and hence sub-optimal decisions with the cube). At least it was a lot better than the 8-65 drubbing that Raccoon received when they reversed the roles of Protagonist/Antagonist. Initially the Raccoon thought the best equaliser for a mediocre player is to play each game at high stakes and hope to get lucky, even if the game state rot13(fhpxrq) since a long match would allow the antagonist to “grind” his way to victory. But the Beaver thought **it was better to be aggressive with even marginal advantages** – for instance if an intermediate player starts with six guaranteed turnovers or a “good five” then he should immediately double. Then at least he is fighting from a position of strength. If the protagonist thought his chances without a doubling cube were 50-50 then he is probably better off grinding and should hope to win on skill, not luck.

And the less said about Ninja Monkey’s first Match-to-65 and his infamous random move algorithm the better 😊