In this month's issue you'll find out how to arrange the longest possible rope on the surface of a sphere and how to use polynomials to solve combinatorial problems. You'll also learn about new approaches to Kirkman's "Fifteen Schoolgirl Problem" and to the theory of nonunique factorization, and much more.
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On Sphere-Filling Ropes
Henryk Gerlach and Heiko von der Mosel
What is the longest rope on the unit sphere? Intuition tells us that the answer to this packing problem depends on the rope’s thickness. For a countably infinite number of prescribed thickness values we construct and classify all solution curves. The simplest ones are similar to the seam lines of a tennis ball; others exhibit a striking resemblance to Turing patterns in chemistry, or to ordered phases of long elastic rods stuffed into spherical shells.
On the Grasshopper Problem with Signed Jumps
The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, was the following. Let a1, a2, . . . , an be distinct positive integers and let M be a set of n - 1 positive integers not containing s = a1 + a2 + . . . + an . A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a1, a2, . . . , anin some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M. In this paper we consider a variant of the IMO problem when the numbers a1, a2, . . . , ancan be negative as well. We find the sharp minimum of the cardinality of the set M which blocks the grasshopper, in terms of n. In contrast with the Olympiad problem where the known solutions are purely combinatorial, for the solution of the modified problem we use the polynomial method.
Kirkman’s Tetrahedron and the Fifteen Schoolgirl Problem
Giovanni Falcone and Marco Pavone
We give a visual construction of two solutions to Kirkman’s fifteen schoolgirl problem by combining the fifteen simplicial elements of a tetrahedron. Furthermore, we show that the two solutions are nonisomorphic by introducing a new combinatorial algorithm. It turns out that the two solutions are precisely the two nonisomorphic arrangements of the 35 projective lines of PG(3, 2) into seven classes of five mutually skew lines. Finally, we show that the two solutions are interchanged by the canonical duality of the projective space.
Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory
Paul Baginski and Scott T. Chapman
Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger’s theorem, Carlitz’s theorem, and Valenza’s theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.
Chaotic Orderings of the Rationals and Reals
Hayri Ardal, Tom Brown, and Veselin JungiÄ‡
In this note we prove that there is a linear ordering of the set of real numbers for which there is no monotonic 3-term arithmetic progression. This answers the question (asked by ErdÅ‘s and Graham) of whether or not every linear ordering of the reals must have a monotonic k-term arithmetic progression for every k.
A Short Proof of Cramér’s Theorem in R
Raphaël Cerf and Pierre Petit
We give a short proof of Cramér’s large deviations theorem based on convex duality. This proof does not resort to the law of large numbers or any other limit theorem.
A Geometric Modulus Principle for Polynomials
We characterize the ascent and descent directions for the modulus of a complex polynomial p(z) at an arbitrary point z0 in the complex plane. We prove that when p(z0) ≠ 0, the cones of ascent and descent directions partition the unit disc centered at z0 into alternating sectors of ascent and descent, each having angle π/k, where k ≥ 1 is the smallest index with p(k)(z0) ≠ 0. Applying this geometric modulus principle, we give new proofs for the maximum modulus principle, the fundamental theorem of algebra, and the Gauss-Lucas theorem.
PROBLEMS AND SOLUTIONS
Corrected version of Problem 11608 (pdf)
Experimental Mathematics in Action, by D. H. Bailey, J. M. Borwein, N. J. Calkin,
R. Girgensohn, D. R. Luke, and V. H. Moll. Reviewed by Andrew Odlyzko.
INDEX TO VOLUME 118
The American Mathematical Monthly Homepage