The Calendar Nim scam from Scam School was actually the 2nd scam idea I submitted to Scam School. This week, the 1st scam idea I submitted to them has been turned into an episode!

OK, to be fair, Penney's Game (or Penney Ante) is well known, and Brian Brushwood does mention that many people sent it in. Note, though, that he does give a special mention to me and the Grey Matters site at the end! Thank you very much, Brian!



James Grime (singingbanana), whom you may remember from the Pi Day Magic Trick, has a great video explanation in two parts. The first part focuses on the rules and how to make the choices, much like the above episode of Scam School. In the second part, he explains the mathematics behind the bet.

If you're interested in more of the technical details, Plus Magazine's most recent issue (at this writing) features an excellent article explaining the math behind Penney's Game.

The part that causes most of the confusion (besides the probabilities) is what it called non-transitivity. It's actually a fancy word for a very simple concept.

Imagine you have 3 cars models, which we'll call A, B, and C. If model A is more expensive than model B, and model B is more expensive than model C, it's not too hard to figure out that model A is more expensive than model C, right? This is a transitive relationship, and is the kind of relationship we're used to when comparing things. Think of transitivity as a relationship where you can rank things in the manner first, second, and so on down to last.

However, this kind of relationship doesn't always hold true. In Rock/Scissors/Paper, for example, rock beats scissors, and scissors beats paper, but this doesn't mean that rock beats paper (as we might expect in a transitive relationship). As we all know, paper beats rock. In this kind of relationship, each item can rank higher than at least one other item in the set, and each item can rank lower than another item in the set. For example, rock beats scissors, but rock can be beaten by paper.

Since this latter type of relationship can't be given a rank like that of a transitive relationship, this second type is called, not surprisingly, non-transitive.

Because we're more familiar with the first (transitive) type of relationship, the qualities of non-transitive relationships can often be quite surprising. Want proof? Since we've used Rock/Scissors/Paper as an example, check out Scam School's use of the game:



If you use specially made dice, such as these or these or the home-made set described in the video below, you can also pull a non-transitive dice scam!



Don't like the idea of special dice for this, the dice could be replaced with 6-card hands using the same numbers, and have cards drawn at random from the players' respective face-down hands. You will need 2 decks to make this up, however, as one hand will contain six 3s.

Getting back to the Penney's Game episode, take another look at Brian's parting joke: “Next week, we're going to be learning, from a memory expert, a mnemonic device that will allow you to memorize the entire constitution in 20 minutes.” Hmmm...should I feel bad that I actually did a series of posts (Part I, Part II, Part III) on this very topic? I do suggest spending more than 20 minutes on it, though.