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.

All meetings take place in

**CM105** unless otherwise indicated.

### Non-transitive games

A non-transitive game is a game for which the various
strategies produce one or more "loops" of preferences. As a
result, in a non-transitive game the fact that strategy A is
preferred over strategy B, and strategy B is preferred over
strategy C, does not necessarily imply that strategy A is
preferred over strategy C. Derren Brown's trick from the video is
based on this idea. Have been extended in various ways,
e.g. to the realm
of quantum games.

### The Kruskal count

The Kruskal Count is a card trick invented by Martin J. Kruskal
in which a magician "guesses" a card selected by a subject
according to a certain counting procedure. With high probability
the magician can correctly "guess" the card. The success of the
trick is based on a mathematical principle related to coupling
methods for Markov chains. See
e.g. arXiv:math/0110143.

### The Diaconis mind-reader

See e.g. p227
of Algebraic
Shift Register Sequences by Goresky and Klapper.

### Hamming codes

Use hamming codes to guess a number with lying allowed, see
e.g. Hamming Codes.

### Other card tricks

See e.g. the card trick from Edwin
Connell's Elements
of Abstract and Linear Algebra, page
18. Or The
best card trick as described by Michael Kleber. There are
many, many variations on this theme, often involving combinatorics
and group theory.

### Fold and One-Cut Theorem

Any straight-line drawing on a sheet of paper may be folded
flat so that, with one straight scissors cut right through the
paper, exactly the drawing falls out, and nothing else. Houdini's
1922 book "Paper Magic" includes instructions on how to cut out a
5-point star with one cut. Martin Gardner posed the general
question in his Scientific American column in 1960. For the
proof, see Chapter 17 of Geometric
Folding Algorithms: Linkages, Origami, Polyhedra. The book includes
instructions for cutting out a turtle.

### Trailing the Dovetail Shuffle to its Lair

Trick based on a careful analysis of the riffle shuffle, in
which an audience member performs a number of riffle shuffles and
then moves a single card, and the magician guesses which card has
been moved. See e.g.
this
text Dave Bayer and Persi Diaconis

### Julian Havil

Has written two books with many examples of non-intuitive
maths Impossible?:
Surprising Solutions to Counterintuitive Conundrums and
Nonplussed!:
Mathematical Proof of Implausible Ideas. Many examples are
simple but some contain substantial math behind them. Also
contains non-statistical examples, e.g. based on graph
theory or geometry.

### Martin Gardner

Key figure in the field of recreational mathematics, who has
written an almost endless supply of articles on surprising
mathematics topics, many of which have appeared in Scientific
American. There is even a Gathering for
Gardner conference dedicated to him.