Penney's Game is a non-transitive probability game involving coin tosses. You invite your friend to name a sequence of coin tosses (at least three long), and you can always name a sequence that is more likely to come up first.
See this link which suggests a card-game variant for a good explanation and a diagram.
Now, what happens if you have five players. Player 1 names HTT as his sequence. Player 2 names HHT in order to beat Player 1. Player 3 names THH to beat Player 1. So Player 4 names TTH to beat Player 3. Player 5 now names HTT to beat Player 4.
If Player 2 usually beats P1, P3 usually beats P2, and P4 usually beats P3, then how does P5 - equivalently P1 usually beat P4? P1 < P2 < P3 < P4 < P1 ??
Bit baffling at first, but I think it's fairly easy to figure out what I've partly neglected to point out.
All this comes from a discussion on the JREF forums of this vos Savant column - which prompts an entirely separate discussion of random strings and probability...
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