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Illusionists such as Derren Brown often make use of the fact that humans have a particularly bad intuition for some of the most basic mathematical facts. His "head or tails" trick for instance, is based around the concept of so-called non-transitive games, a well-studied topic in mathematics. Another one is the famous "three door problem" (or Monty Hall problem).

Apart from their use by illusionists, counterintuitive examples such as these are often also extremely useful for teaching purposes (much more fun than boring examples which just confirm what you already "knew" anyway).

In this project you will study some of this counterintuitive mathematics and applications of it to real world problems or real world entertainment.

Depends on the particular sub-topic you decide to work on.

There is plenty of material in print and on the web that discusses the use of counterintuitive mathematics, see for instance

- Michael Kleber, "The best card trick".
- Julian Havil, "Impossible?: Surprising Solutions to Counterintuitive Conundrums" and "Nonplussed!: Mathematical Proof of Implausible Ideas"
- Wallace Lee, "Math Miracles".