In this issue, Jacob Siehler shows us how to win Conway's M13 solitaire game, William E. Wood introduces us to Squigonometry, and Jonathan Hodge tells us about the mathematics of referendum elections. We watch Cinderella and her Stepmother play a game with overflowing buckets, and we see the problems and solutions from this summer's USAMO, USAJMO, and IMO.—Walter Stromquist, Editor
Depth and Symmetry in Conway’s M13 Puzzle
We analyze a sliding tile puzzle (due to J. H. Conway) which gives a presentation of the Mathieu group M12, one of the sporadic simple groups. We use the symmetries of a finite projective plane to classify automorphisms of M12 which preserve the depth of puzzle positions, and give an algorithm for solving the puzzle by hand.
William E. Wood
Differential equations offers one approach to defining the classical trigonometric functions sine and cosine that parameterize the unit circle. In this article, we adapt this approach to develop analogous functions that parameterize the unit squircle defined by x4 + y4 = 1. As we develop our new theory of “squigonometry” using only elementary calculus, we will catch glimpses of some very interesting and deep ideas in elliptic integrals, non-euclidean geometry, number theory, and complex analysis.
The Editor’s Song
Frank A. Farris
The program of the 2011 Mathfest’s opening banquet was “MAA—The Musical!” Produced by Annalisa Crannell and starring the “MAA Players” (active MAA members all), it highlighted activities of the Association and of Mathfest itself. This song represents the journals. It was sung by past editor Frank Farris to the tune of “A Wand’ring Minstrel I,” from Gilbert and Sullivan’s the Mikado.
The Mathematics of Referendum Elections and Separable Preferences
Jonathan K. Hodge
Voters in referendum elections are often required to cast simultaneous ballots on several possibly related questions or proposals. The separability problem occurs when a voter’s preferences on one question or set of questions depend on the known or predicted outcomes of other questions. Nonseparable preferences can lead to seemingly paradoxical election outcomes, such as a winning outcome that is the last choice of every voter. In this article, we survey recent mathematical results related to the separability problem in referendum elections. We explore the structure of interdependent preferences, consider related combinatorial and algebraic results, and examine the practical impact of separability on the outcomes of referendum elections.
How Cinderella Won the Bucket Game (and Lived Happily Ever After)
Antonius J.C. Hurkens, Cor A.J. Hurkens, and Gerhard J. Woeginger
The paper investigates a combinatorial two-player game, in which one player (Cinderella) wants to prevent overflows in a system with five water-buckets whereas the other player (the wicked Stepmother) wants to cause such overflows. Several sub-optimal and optimal strategies for both players are analyzed in detail.
Edge Tessellations and Stamp Folding Puzzles
Matthew Kirby and Ronald Umble
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. In this article we prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon. A stamp folding puzzle is a paper folding problem constrained to the perforations on a sheet of postage stamps. Such sheets necessarily embed in an edge tessellation. On page 143 of his book Piano-Hinged Dissections: Time to Fold!, G. Frederickson poses the following conjecture: “Although triangular stamps have come in a variety of different triangular shapes, only three shapes seem suitable for [stamp] folding puzzles: equilateral, isosceles right triangles, and 60º-right triangles.” We prove that the four non-obtuse polygons mentioned above generate edge tessellations suitable for stamp folding puzzles. Our proof of suitability, which establishes Frederickson’s Conjecture, exhibits explicit algorithms for folding each suitable edge tessellation into a packet of single stamps.
Folding Noneuclidean Strips of Paper
Nikolai A. Krylov and Edwin L. Rogers
We have considered the process of inscribing triangles between parallel lines in the Euclidean plane using an iterative process. It is a familiar result that the limiting triangle is equilateral. In this Note, we demonstrate that the same result holds when the geometry is hyperbolic and the lines are asymptotically parallel. We further extend the process to the sphere, where it is found that the limiting triangle is isosceles.
Proof Without Words: Fibonacci Trapezoids
Hans R. Walser