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Vol. 39, No. 2, pp. 90-170

ARTICLES

**Universal Stoppers are Rupert**

Richard P. Jerrard and John E. Wetzel

A stopper is called *universal* if it can be used to plug pipes whose cross-sections are a circle, a square, and an isosceles triangle, with the diameter of the circle, the side of the square, and the base and altitude of the triangle all equal. Echoing the well-known result for equal cubes that is attributed to Prince Rupert, we show that it is always possible to make a hole in such a universal stopper that is large enough to permit the passage of a second such stopper.

**Mind the Gap**

Thomas J. Bannon and Robert E. Bradley

If you break a stick at two random places, the probability that the three pieces form a triangle is ¼ . How does this generalize? To answer this question, we give a method for finding the probability that n randomly chosen points in a given interval fall within a specified distance of one another. We use this method to provide solutions to generalized versions of three popular probability problems.

**Finding All Solutions to the Magic Hexagram**

Jason Holland and Alexander Karabegov

In this article, a systematic approach is given for solving a magic star puzzle that usually is accomplished by trial and error or "brute force." A connection is made to the symmetries of a cube, thus the name Magic Hexahedron.

**A New Property of Repeating Decimals**

Jane Arledge and Sarah Tekansik

As extended by Ginsberg, Midi's theorem says that if the repeated section of a decimal expansion of a prime is split into appropriate blocks and these are added, the result is a string of nines. We show that if the expansion of 1/*p*^{n}+1 is treated the same way, instead of being a string of nines, the sum is related to the period of 1/*p*^{n}.

**Fibonacci's Forgotten Number**

Ezra Brown and Cornelius Brunson

Fibonacci's forgotten number is the sexagesimal number 1;22,7,42,33,4,40, which he described in 1225 as an approximation to the real root of *x*^{3} + 2*x*^{2} + 10*x* - 20. In decimal notation, this is 1.36880810785... and it is correct to nine decimal digits. Fibonacci did not reveal his method. How did he do it? There is also a curious mistake in his answer: why is it there? We first describe how Leonardo came to know this number. We then introduce several methods that he may have used for approximating roots of polynomials. Finally, we make a guess as to how he really did it.

**Two Problems with Table Saws**

William R. Vautaw

We solve two problems that arise when constructing picture frames using only a table saw. First, to cut a cove running the length of a board (given the width of the cove and the angle the cove makes with the face of the board) we calculate the height of the blade and the angle the board should be turned as it is passed over the blade. Second, to construct a rectangular frame whose outer edges will slant out from a wall, we calculate the angles at which the ends of the pieces must be mitered and beveled. The solutions use analytic geometry, trigonometry, vector analysis, and differential calculus.

**Remainder Wheels and Group Theory**

Lawrence Brenton

Why should prospective elementary and high school teachers study group theory in college? This paper examines applications of abstract algebra to the familiar algorithm for converting fractions to repeating decimals, revealing ideas of surprising substance beneath an innocent façade.

**CAPSULES**

**On the number of trailing zeros in ***n*!

David S. Hart, James E. Marengo, Darren A. Narayan, and David S. Ross

There are two trailing zeros at the end of 10!, 3,628,800. There are 24 trailing zeros at the end of 100!, 1000! has 249, and 10,000! has 2,499. Do you see a pattern? Look closely for another geometric series!

**The Naïve Chain Rule**

M. Leigh Lunsford, Marcus Pendergrass, Phillip Poplin, and David Shoenthal

Calculus instructors are familiar with "logic" like this:

. The authors seek to find functions *f*and *g* for which the "naïve chain rule" holds. Along the way they encounter some interesting mathematics, including ideas from elementary dynamical systems.

**The Naïve Product Rule for Derivatives**

Carter C. Gay, Akalu Tefera, and Aklilu Zeleke

Two functions *f* and *g* are said to satisfy the naïve product rule if (*fg*)' = *f'g*'. In this note we go a step further and look for pairs of functions that satisfy the corresponding rule for second derivatives, (*fg*)*"* = *f"g"*.

**The Right Theta**

William Freed and Athanasios Tavouktsoglou

The formula *θ* = arctan (*y/x*) gives the angle associated with a point (*x,y*) in the plane, valid for |*θ*| θ|