Given a positive integer \(m\), the authors exhibit a group with the...

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**Zigzags**

Peter J. Giblin

A 'zigzag' is constructed in the plane by a process involving the choice of two lengths and two angles. The 'zags' (alternate sides of the zigzag) are then extended to infinite lines and the zigs (remaining sides of the zigzag) are deleted. This gives a finite family of lines in the plane, which often has a striking envelope: that is, there is often a visually evident curve which is tangent to all these lines. Sometimes the envelope consists of several overlapping congruent curves. Selecting one component of the envelope, the problem is to identify precisely this visually evident curve. Choosing an arbitrary curve tangent to all the lines does not work: the result is not necessarily visually correct. Two methods are given for identifying the visually correct curve, both related to a construction called a 'whirligig' where a line is carried by a spinning circle whose centre moves round another circle. Many examples are given, including some in which the components of the envelope are all circles, in which case it might be quite hard to tell just how many circles there are.

**Four Constants in Four 4s**

A. Bliss, S. Haas, J. Rouse, G. Thatte

We approximate four mathematical and physical constants using four fours.

**The Track of a Bicycle Back Tire**

Steven R. Dunbar, Reinier J. C. Bosman, and Sander E. M. Nooij

A rider on a bicycle goes down a road, the front tire steering a path by following a prescribed motion, or just weaving back and forth. We can see from the tire tracks after passing through a puddle that the back tire follows a path similar to the path of the front tire. If we know the path of the front tire precisely, what is the path of the back tire? In this article we derive and solve differential equations for the path of the back tire, knowing the path of the front tire. For a parametric form of the front tire path the differential equations for the path of the back tire are a pair of coupled nonlinear differential equations. We can solve the differential equations directly in some cases, with geometrical arguments, with ``guess-and-check'', and with perturbation methods, i.e., Taylor expansion methods, in other cases.

**Volumes and Cross-Sectional Areas**

William T. England and T. Len. Miller

In elementary calculus, we calculate the volume of a solid by integrating the area of cross sections of the solid orthogonal to one of the axes. In this note we calculate volumes by integrating the area of cross sections of the solid orthogonal to a curve.

**Two Reflected Analyses of Lights Out**

Oscar Martin-Sanchez and Cristobal Pareja-Flores

Two analyses of Lights Out are shown each in a column: the left one addresses a person who is interested and methodical; the right one, a mathematician. These analyses are mirror images of one another: the concepts, examples, and figures on each side are designed to enrich their counterparts. A parallel reading of the two columns is recommended. Our aim is to provide the reader with an understanding of the game, efficient algorithms to know when the game can be solved, and also how to find the solution. Our work offers a way to solve the puzzle with the game in hand, without needing a computer or even pencil and paper.

**An Equilateral Triangle with Sides Through the Vertices of an Isosceles Triangle**

Fukuzo Suzuki

There are many interesting problems in traditional Japanese Mathematics, known as "Wasan." A problem proposed by T. Sakuma (1819-1896) is one of them. It is a result establishing the fixed relation among certain lengths that arise when an equilateral triangle is drawn with sides passing through the vertices of an isosceles triangle. This problem is solved in full generality and another problem of Wasan is posed for the reader.

**The Arithmetic-Geometric Mean Inequality and the Constant e**

Hansheng Yang and Heng Yang

Existence of constant e and concerning issues are always attractive to some mathematicians. The work made by T.N.T. Goodman and C.W. B. Barnes is surprising; however, a question occurs Â– is there any easier and more elementary proof even strong result? From two different perspectives, we do the proof successfully by using the well-known arithmetic-geometric mean inequality.

**Two Irrational Numbers that Give the Last Non Zero Digits of n! and n^n**

Gregory P. Dresden

Imagine an infinite decimal whose nth digit is the last (i.e., right most) non zero digit of n!. We show that this decimal, although exhibiting unusual patterns, is in fact irrational, and we show the same for n^n in place of n!.

**Cramer’s Rule Is Due to Cramer**

Antoni A. Kosinski

The note documents that, contrary to the often expressed opinion, it was Cramer, not Maclaurin, who first stated Cramer’s rule correctly and in full generality.

**Dialog with Computer in the Proof of the Four-Color Theorem **

Jasper Memory

A poem in which the computer repeatedly assures the user that all one needs are Â“Red, green, blue, yellow.Â”

**PROOFS WITHOUT WORDS**

**Equilateral Triangle**

James Tanton

For an equilateral triangle, the sum of the distances from any interior point to the three sides equals the height of the triangle.

**How Did Archimedes Sum Squares in the Sand?**

Katherine Kanim

This Proof Without Words illustrates every step of Archimedes’s proof of his formula for a sum squares.

The 2000 VIEWPOINTS Group

The formula for the sum of the geometric series is illustrated by a diagram in which Picard iteration is used to find the fixed point of the function y = r x + a.

edited by Elgin H. Johnston

**REVIEWS edited by Paul J. Campbell**