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MONTHLY, April 2004

**Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes**

by Roe Goodman

goodman@math.rutgers.edu

Let us imagine Lewis Carroll’s Alice in her chamber in front of a peculiar cone-shaped arrangement of three looking glasses. She steps through one of the looking glasses and finds that she is in a new *virtual chamber*. Passing through one of the *virtual mirrors*, she continues her exploration of many virtual chambers until she suddenly finds herself back in her own real chamber, just in time for tea. Eager to have new adventures, Alice wonders how many different ways the mirrors could be arranged so that she could have other trips through the looking glasses and still return some day for tea. Alice’s problem was solved (for all dimensions) by H.S.M. Coxeter in 1934, who classified all possible systems of *n* mirrors through 0 in *n*-dimensional Euclidean space whose reflections generate a finite group of orthogonal matrices. I describe Coxeter’s results, emphasizing the connection with kaleidoscopes. I also give plans for three-dimensional kaleidoscopes that I have built which exhibit the symmetries of the Platonic solids in an intriguing way. http://math.rutgers.edu/~goodman

**Stochastic Apportionment**

by Geoffrey Grimmett

grg@dpmms.cam.ac.uk

Ten goats are to be assigned to three brothers in numbers proportional to the ages (in years) of the recipients. Given the integral nature of a goat, it is not generally possible to meet exactly the condition of proportionality, and the resulting "apportionment problem" is a classic of operational research. The most studied and contentious instance of the problem is the allocation to states of the 435 seats in the U.S. House of Representatives. We propose an apparently new method of apportionment that is stochastic, that meets the so-called quota condition, and that is fair in the sense of mean values. Two sources of systematic unfairness are identified, firstly the lower bound condition (every state shall receive at least one seat), and secondly the lower quota condition (every state shall receive at least the integral part of its quota).

**Evaluation of Ill-Behaved Power Series**

by Pirooz Mohazzabi and Thomas A. Fournelle

pirooz.mohazzabi@uwp.edu, thomas.fournelle@uwp.edu

Ill-behaved power series of mathematical physics that pose insurmountable computational difficulties and require unreasonable high precisions and computing resources may be transformed into linear differential equations that can be solved numerically in a straightforward fashion. Several examples are discussed.

**Fermat’s Last Theorem for Rational Exponents**

by Curtis D. Bennett, A.M.W. Glass, and Gábor J. Szekely

cbennett@lmu.edu, A.M.W.Glass@dpmms.cam.ac.uk, gabors@bgnet.bgsu.edu

Fermat’s Last Theorem states that *a*^{n} + *b*^{n} = *c*^{n} has no positive integer solutions for *n* > 2. How does allowing for rational exponents with numerator greater than 2 and complex roots change the conclusion? Surprisingly, classical triangles and the laws of sines and cosines become embroiled in the proof that the only new solutions arise from taking sixth roots of unity and *a, b,* and *c* of equal modulus.

**Two Exams Taken by Ramanujan in India**

by Bruce C. Berndt and C. A. Reddi

According to Ramanujan’s biographers, he took only three exams outside of primary and secondary schools. First, Ramanujan sat for the Matriculation Exam to enter the Government College of Kumbakonam. Second, he took an exam at the end of his first and only year at the Government College. Third, he took the First Arts Examination of the University of Madras in an attempt to obtain a degree by examination. Searching in the Tamil Nadu Archives, the second author found copies of the first and third exams. We reproduce these exams in our article and offer extensive comments on Ramanujan’s performances.

**Notes**

**The Wave Equation, Mixed Partial Derivatives, and Fubini’s Theorem**

by Asuman Aksoy and Mario Martelli

asuman.aksoy@claremontmckenna.edu, mario.martelli@claremontmckenna.edu

**One Observation behind Two-Envelope Puzzles**

by Dov Samet, Iddo Samet, and David Schmeidler

samet@post.tau.ac.il, iddo.samet@cash-u.com, schmeid@post.tau.ac.il

**On the "Reducibility" of Arctangents of Integers**

by E. Kowalski

emmanuel.kowalski@math.u-bordeaux.fr

**Kepler’s First LawÂ—A Remark**

by Paul Monsky

Monsky@Brandeis.edu

**An Elementary Proof of Krull’s Intersection Theorem**

by Hervé Perdry

herve@matesco.unican.es

**Lattices in C and Finite Subsets of a Circle**

by Jacob Mostovoy

jacob@matcuer.unam.mx

**Problems and Solutions**

**Reviews**

**Advanced Modern Algebra**

by Joseph J. Rotman

Reviewed by Daniel R. Farkas

farkas@math.vt.edu

**Practical Analysis in One Variable **

by Donald Estep

Reviewed by Paul Zorn

zorn@stolaf.edu