*A Fair-Bold Gambling Function Is Simply Singular* by Richard D. Neidinger and* The Looping Rate and Sandpile Density* *of Planar Graphs* by Adrien Kassel and David B.Wilson. Good things often come in short notes - check out *A Function That Is Surjective on Every Interval* by David G. Radcliffe. Darren Glass reviews *The Fascinating World of Graph Theory *by Arthur Benjamin, Gary Chartrand, and Ping Zhang. And if all that is not enough, don't forget our award winning Problem Section. Stay tuned for the February Monthly when we will learn whether or not we can play a fair game of craps with a loaded pair of dice. —*Scott T. Chapman, Editor*

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## Table of Contents

## Articles

### A Fair-Bold Gambling Function Is Simply Singular

Richard D. Neidinger

A new variation on a classic singular function has derivative values that, unlike the classic, can be simply characterized by secant slopes between dyadic rationals. A singular function is defined to have derivative zero almost everywhere, even though it is continuous and (in this case, strictly) increasing. Points where the derivative of the new variation is zero and points where it is infinite are characterized in terms of binary digits. The classic function is described as the probability of reaching a goal when gambling on a fixed-probability game; the variation uses an alternating probability.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.3

### The Looping Rate and Sandpile Density of Planar Graphs

Adrien Kassel and David B. Wilson

We give a simple formula for the looping rate of a loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is well suited to taking limits where the graph tends to an infinite lattice, and we use it to give an elementary derivation of the (previously computed) looping rate and sandpile densities of the square, triangular, and honeycomb lattices, and compute (for the first time) the looping rate and sandpile densities of many other lattices, such as the kagomé lattice, the dice lattice, the truncated hexagonal lattice (for which the values are all rational), and the square-octagon lattice (for which it is transcendental).

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.19

### Integration by Parts and by Substitution Unified with Applications to Green's Theorem and Uniqueness for ODEs

J. Ángel Cid and Rodrigo López Pouso

We present a rather unknown version of the change of variables formula for nonautonomous functions. We will show that this formula is equivalent to Green's theorem for regions of the plane bounded by the graphs of two continuously differentiable functions. Moreover, the formula has interesting applications in the uniqueness of solution of ordinary differential equations.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.40

### A Generalization of Euler's Theorem for *ζ*(2*k*)

M. Ram Murty and Chester Weatherby

We extend Euler's celebrated theorem evaluating *ζ*(2*k*). We replace the terms *n*−2*k* in the infinite sum for *ζ*(2*k*), with (*n*2 + *Bn* + *C*)−*k* where *B*, *C* are complex and *k* is a positive integer. We explicitly evaluate these sums and also briefly discuss their transcendence.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.53

### Beyond Schur's Conjecture

Shivarajkumar

Schur [**4**] conjectured that the maximum length *N* of consecutive quadratic nonresidues modulo a prime *p* is less thanif *p* is large enough. This was proved by Hummel in 2003. In this note, we outline a clear improvement over Hummel's bound for *p* > 23.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.66

### The Diameter of Chess Grows Exponentially

Yaroslav Shitov

We present an infinite sequence of pairs (*A**n*, *B**n*) of chess positions on an *n* × *n* board such that (1) there is a legal sequence of chess moves leading from *A**n* to *B**n* and (2) any legal sequence leading from *A**n* to *B**n* contains at least exp(*n* + *o*(*n*)) moves.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.71

## Notes

### On a Property Motivated by Groups with a Specified Number of Subgroups

Michael C. Slattery

We consider finite groups with a given number of subgroups. With a carefully chosen definition of “similar,” we find that for each positive integer k, the groups with exactly k subgroups fall into a finite list of similarity classes.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.78

### Pictorial Calculus for Isometries

Oleg Viro

In this paper a new graphical calculus for operating with isometries of low dimensional spaces is proposed. It generalizes a well-known graphical representation of vectors and translations in an affine space. Instead of arrows, we use arrows framed with affine subspaces at their end points. The head to tail addition of vectors and translations is generalized to head to tail composition rules for isometries. The material of this paper is elementary and can be used even in the framework of high-school geometry.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.82

### A Function That Is Surjective on Every Interval

David G. Radcliffe

We exhibit a real function that is surjective when restricted to any nonempty open interval.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.88

### Iterations for the Lemniscate Constant Resembling the Archimedean Algorithm for Pi

Thomas Osler

We give an iterative algorithm that converges to the lemniscate constant *L*. This algorithm resembles the famous Archimedean algorithm for *π*. The derivation is based on the recently discovered product of nested radicals forby Aaron Levin. Levin's product closely resembles Vieta's historic product for.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.90

### A Probabilistic Proof of the Multinomial Theorem

Kuldeep Kumar Kataria

In this note we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic probability concepts.

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.94

## Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.97

## Book Review

*The Fascinating World of Graph Theory *Arthur Benjamin, Gary Chartrand, and Ping Zhang

Reviewed by Darren B. Glass

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.1.106

## MathBits

### Monthly Gems

Daniel J. Velleman

### A Partial Solution of Representing 1 as a Sum of *n* Distinct Unit Fractions

Konstantinos Gaitanas

### A Synthetic Approach to a USAMO Problem

Xiaoxue Li

### A Note on the Logarithmic Mean

József Sándor

### 100 Years Ago This Month in the American Mathematical Monthly

Edited by Vadim Ponomarenko