# Mathematics Magazine - April 2004

### ARTICLES

The Fabulous (11,5,2) Biplane
Ezra (Bud) Brown
87-100
A question by a graduate student in sociology led to this story about a combinatorial object, its symmetries, and its many connections to the worlds of block designs, coding theory, and finite simple groups. We investigate these connections (Steiner systems, Golay codes, Mathieu groups) and get to know this object, commonly known as the (11,5,2) biplane, quite well.

Perfect Shuffles through Dynamical Systems
Daniel J. Scully
101-117
To perform a perfect shuffle on a deck of 2n cards you split the deck into two packets of n cards each and interleave the cards in the two packets together perfectly. Magicians do perform perfect shuffles. They appeal to magicians because they appear random but are not. In this article we present a new way to model perfect shuffles that uses the well-known doubling function from dynamical systems. This model shows the intimate relationship between the orbits of the cards under the perfect-shuffle permutations and the binary expansions of certain rational numbers from the unit interval. We exploit that relationship to show connections between the various perfect shuffles on the various sized decks, and we use it to link deck sizes to orbit lengths. The model generalizes nicely to k-handed perfect shuffles, and we use the model to describe more general (less-than-perfect) riffle shuffles.

The Geometry of Generalized Complex Numbers
Anthony A. Harkin and Joseph B. Harkin
118-129
It may come as a surprise to some that there are useful definitions of the imaginary unit i besides i2 = -1. The alternative definitions i2=1 and i2 = 0 give rise to complex number systems first studied by W. Clifford and E. Study. What are the properties of these complex number systems? The geometries of the complex planes corresponding to these modified definitions of the imaginary unit turn out to be related to the geometries of Minkowski and Laguerre.

### NOTES

Emeric Deutsch and Warren P. Johnson
130-134
Given a property that a permutation may or may not have, we define a statistic on all permutations: the length of the longest initial segment of the permutation that does have the given property.

### PROOF WITHOUT WORDS

Four Squares with Constant Area
Roger B. Nelsen
135
A proof without words of the following statement: If two chords in a circle intersect at right angles, then the sum of the squares of the lengths of the four segments formed is a constant (the square of the length of the diameter).

Inversions and Major Index for Permutations
Thotsaporn Thanatipanonda
136-140
A bijective proof of the equidistribution of two permutation statistics—inversion number and major index —was given by Foata in 1968. In this paper, the author gives an elementary bijective proof of the equidistribution of these two statistics for the permutations of the set {1,2,...,n}.

Permutations in the Sand
Mark D. Schlatter
140-145
We look at a class of sona drawings (a form of African art) and determine when these drawings can be traced without lifting your finger. We use the idea of mirror curves to create the drawings and analyze them using permutations.

The Minimal Polynomials of sin (2π/p) and cos (2π/p)
Scott Beslin and Valerio De Angelis
146-149
If p is an integer, sin (2π/p) and cos (2π/p) are algebraic numbers, that is, they are roots of some irreducible polynomial with integer coefficients, called the minimal polynomial. In this note, we give an elementary derivation of such polynomials in the case that p is prime number, using only Eisenstein irreducibility criterion.

Final Digit Strings of Cubes
Daniel P. Biebighauser, John Bullock, and Gerald A. Heuer
149-155
A problem in the North Central Section/MAA Team Contest of November, 2000, asked whether there is an integer, the cube of which (in decimal form) ends in 2000 ones. The problem is interesting because the answer is "yes,Â” and naturally raises the question of exactly what strings of digits occur as final digit strings of cubes. It turns out that if the last digit in the string is 1, 3, 7, or 9, there is always a cube ending in this string of digits. If the last digit is something else, there may or may not be such a cube, and this note gives the exact conditions on the digit string under which such a cube exists.