There are a set of special dice, called Efron's Dice. Each die will win over the last. And it will keep going until there's an ultimate winner - which will lose to the original loser. There's a lesson, there, that frustrates many sports fans. Here's how to snatch victory from the jaws (or even the stomach) of defeat.

Efron's Dice

Let us play a little game with a set of Efron's Dice. The game consists of a series of throws, one die against another, with the pot compounding the whole time. Efron's Dice aren't quite like others. They are a standard six-sided shape, but the first match with be between one die with faces that read 5, 5, 5, 1, 1, 1, and one die that reads, 6, 6, 2, 2, 2, 2. Despite the seeming advantage of that extra five, the second die has a probability of winning of two-thirds.

The winner moves on! The die reading 6, 6, 2, 2, 2, 2, is now up against a die reading 3, 3, 3, 3, 3, 3. This new die has a probability of winning of two-thirds, so it will, in most matches, overthrow the previous champion and move on.

Next up! The all-threes die will go up against a die reading 4, 4, 4, 4, 0, 0. This new die has a probability of winning of two-thirds.

Now for the final match. This winning die, this die that came out on top, will go up against the die that got beaten in the first match. The champion 4, 4, 4, 4, 0, 0, will go up against old, unlucky 5, 5, 5, 1, 1, 1. A quick look at the comparative sides will tell you what's likely to happen. That disgraced losing die now has a probability of winning of two-thirds.

That's the trick of Efron's Dice. Put in the right sequence of steps, and every die has a two-thirds chance of winning out over the last die, until you get to the end of the sequence and find out that the highest "step" is suddenly lower than the lowest one.

The Problem With Play-Offs

The trick is to stop seeing the odds as steps, or as terms of absolute odds, or anything else, because those ideas are transitive. Objects have a transitive relationship if there is a kind of linear progression between them. A dog is bigger than a mouse. Because size is (for the most part) a transitive relation, we can confidently say that anything bigger than a dog is also bigger than a mouse. Likewise, anything smaller than a mouse is also smaller than a dog.