Stan Wagon

Macalester College · Department of Mathematics, Statistics, and Computer Science

Considering regions in a map to be adjacent when they have nonempty intersection (as opposed to the traditional view requiring intersection in a linear segment) leads to the concept of a facially complete graph: a plane graph that becomes complete when edges are added between every two vertices that lie on a face. Here we present a complete catalog...

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to find...

We present a result of Mycielski and Sierpiński—remarkable and underappreciated in our view—showing that the natural way of eliminating the Banach–Tarski paradox by assuming all sets of reals to be Lebesgue measurable leads to another paradox about division of sets that is just as unsettling as the paradox being eliminated. The division paradox ass...

We investigate the problem of finding integers $k$ such that appending any number of copies of the base-ten digit $d$ to $k$ yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending {\it any} digit yields a composite number.

A puzzle about prisoners trying to identify the color of a hat on their head leads to a version that is related to the independence number of the arrangement graph A(m,n), to linear fractional transformations in finite fields, and to Steiner systems. We present several results, the main ones counterexamples to the natural conjecture that perfect ha...

Consider this classic puzzle: Charlie runs four miles at A miles per hour and then four miles at B mph. What is his average speed for the eight miles?

We investigate the following puzzle and several variations, some of which have quite surprising answers. Alice and Bob are in prison under the care of warden Charlie. Alice will be brought into Charlie’s office and shown 52 cards, face-up in a row in an arbitrary order. Alice can interchange two cards. Charlie then turns all cards face-down in thei...

In 1958, S. \'Swierczkowski proved that there cannot be a closed loop of congruent interior-disjoint regular tetrahedra that meet face-to-face. Such closed loops do exist for the other four regular polyhedra. It has been conjectured that, for any positive \epsilon, there is a tetrahedral loop such that its difference from a closed loop is less than...

Paper 13: Stanley Rabinowitz and Stan Wagon, “A spigot algorithm for the digits of π,” American Mathematical Monthly, vol. 102 (March 1995), p. 195–203. Copyright 1995 Mathematical Association of America. All Rights Reserved. Synopsis: In this paper, Rabinowitz and Wagon introduce a very interesting “spigot algorithm” for the digits of π. In partic...

Paper 6: Stan Wagon, “Is pi normal,” The Mathematical Intelligencer, vol. 7 (1985), p. 65–67. With permission of Springer. Synopsis: As our earlier Compendium makes clear, mathematicians have been fascinated by the decimal expansion (and expansions in other bases) of π since the time of Archimedes — what sort of number is π? Questions such as wheth...

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that...

We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that...

The Banach-Tarski Paradox is a most striking mathematical construction: it asserts that a solid ball can be taken apart into finitely many pieces that can be rearranged using rigid motions to form a ball twice as large. This volume explores the consequences of the paradox for measure theory and its connections with group theory, geometry, set theor...

A bipartite graph is Hamilton-laceable if for any two vertices in different parts there is a Hamiltonian path from one to the other. Using two main ideas (an algorithm for finding Hamiltonian paths and a decomposition lemma to move from smaller cases to larger) we show that the graph of knight's moves on an m x n board is Hamilton-laceable iff m >=...

We describe the solution to a combinatorial optimization problem that arises in higher education: the assignment of first-year students to introductory seminars. We trace the evolution of our implementation at Macalester College, a small liberal arts school. We first describe how the classic Assignment Problem for bipartite graphs can be used. Then...

Longtime problem poser Stan Wagon shares his most surprising problems.

Chess pieces of various sorts give rise to intriguing graphs and studying their properties can yield nice conjectures, and sometimes simple proofs. This paper examines some problems related to traditional queens and bishops, and also some pieces arising in a hexagonal version of chess. Using powerful algorithmic methods such as integer-linear progr...

Two travelers arrive at Hilbert's Grand Hotel. The first asks for a room. Hilbert asks, "Smoking or nonsmoking?" "I need a smoking room," he replies. " You're in luck, sir; we are full, but we can move everyone to an adjacent room, which will free up a smoking room." The second traveler then asks for a nonsmoking room. " Sorry, sir, that's not poss...

We investigate the problem of finding integers k such that appending any number of copies of the base-ten digit d to k yields a composite number. In particular, we prove that there exist infinitely many integers coprime to all digits such that repeatedly appending any digit yields a composite number.

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, surprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

A classic textbook problem is to show, assuming Newton’s law of cooling, that if cold milk is added to coffee that has been cooling down, the result will be colder than if the milk was added at an earlier time. We formulate and prove a theorem that shows this holds when the linear function of Newton’s law is replaced by any function satisfying a ce...

A notorious interview question for Facebook job candidates sends the authors off to the races.

Then a computer with an attached camera points at an external display showing the current screen, one sees a familiar visual feedback loop related to the iteration of an affine function. Of course, the same effect occurs when one looks into a mirror that faces another mirror; this is in essence the same as the familiar audio feedback that occurs wh...

We solve the problem of designing a simple device that uses rotary motion to drill a hole with a cross-section that is a regular polygon with an odd number of sides: the main idea is to use a polygonal trammel and a family of rotors. By using different rotors, one can produce a hole with a cross-section that is in any proportion of the trammel size...

The continuous tools from linear programming are brought to bear on the traveling salesman problem and other discrete variants — with cycle-smashing success.

The snail ball is a device that rolls down an inclined plane, but very slowly, repeatedly coming to a stop and staying motionless for several seconds. The interior of the ball is hollow, with a smaller solid ball inside it, surrounded by a very viscous fluid. We show how to model the stop-and-start motion by analyzing the cycloidal curve that would...

Suppose Alice has a coin with heads probability $q$ and Bob has one with heads probability $p>q$. Now each of them will toss their coin $n$ times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given $p,q$, what is the choice of $n$ that maximizes Alice's chances of winning? The problem of deter...

In this third edition of Mathematica® in Action, award-winning author Stan Wagon guides beginner and veteran users alike through Mathematica's powerful tools for mathematical exploration. The transition to Mathematica 7 is made smooth with plenty of examples and case studies that utilize Mathematica's newest tools, such as dynamic manipulations and...

There are several ways to examine the dynamics of the quadratic map defined by f r (x) = rx(1−x). The image above is from a demonstration that shows the bifurcation diagram for f r , while allowing the user to move a slider (the red line) which causes the corresponding cobweb plot to appear as well. The example shown has r set to 3.84, for which th...

Placing wheels on wheels on wheels and giving them different rates of spin leads to some interesting parametric plots. The images show four examples. They arise from the values below, clockwise from upper left, as explained in §1.7.

A space-filling curve is a continuous function P from the unit interval onto the square. The upper images show approximations to the image of P in the square. Further approximations would yield a fully black square. The three-dimensional image shows the true graph of the function: the subset of ℝ3 given by {(t, x, y): P(t) = (x, y)}.

Often a careful look at an old idea from a modern point of view can lead to some interesting developments. In this chapter, we use Mathematica to illustrate several aspects of the four-color theorem, for both maps and graphs. As a consequence, we obtain a randomized algorithm based on Kempe’s 1879 “proof” of the four-color theorem. The algorithm se...

Almost all institutions that rely in a serious way on serial numbers use a check digit scheme to enhance the number and maximize the chance that a computer can detect an error when the number is input. The United States Postal Service, UPS, FedEx, airlines, credit card companies, grocery stores, blood banks, money banks, and driver’s license bureau...

The primes in the Gaussian integers — numbers of the form a + bi — are a fascinating object of study. For example, 2 is not prime as it factors as (1 + i) (1 − i); 3 remains prime. One can ask how far one can walk in the Gaussian primes starting near the origin at 1 + i and taking steps of size no greater than k. The cover image shows how far one c...

The Costa surface is a relatively new type of minimal surface, and it is topologically equivalent to a torus with three punctures. The images shown are from an animation that uses a very simple morphing idea to go from a torus to the Costa surface.

Many famous mathematicians have found relatively simple functions that model the behavior of π(x), the number of primes below x. This graph shows the error, up to one million, of three such approximations: Legendre and Chebyshev used logarithms (blue graph; beyond 1012, Chebyshev is better than Legendre), Gauss (red) used the integral of the recipr...

The diagram illustrates the first eight and the next-to-last moves of a very surprising puzzle. Take a round cake with icing only on the top. Cut out a piece making an angle of 1 radian at the center, turn it upside-down, and reinsert it into the cake. Then move 1 radian clockwise and do the same with a second piece. Continue this process in the cl...

When the construction of the classical Cantor set is extended to the complex plane, many remarkable images result, such as these four (see §6.3).

Can a room in 3-space be designed so that a small person can find a hiding place that is invisible from guards located at every vertex of the room? Such a two-dimensional polygon cannot exist. But the image shows a three-dimensional room that does contain such a hiding place. The roof of the room has been removed so we can see inside: there are six...

A damped pendulum can be described by the differential equation x″ = −x ′−10sinx, where x represents the angular displacement and −x′ is the damping term. The image shows 24 solutions in the phase plane; the solutions almost always converge to one of the equilibrium points, the exception being the separatrix curves. The image also shows the equilib...

Part of a plane tiling using Penrose rhomb tiles: they tile the plane nonperiodically and cannot be used to tile the plane periodically. The first example of such an aperiodic set had over 20 000 types of tiles. Using replacements to implement the recursive ideas that underlie a Penrose tiling makes the generation of such images in Mathematica simp...

Symbolic computations can lead to unexpected and new formulas for famous constants. The four formulas displayed here (from [AW]) all arose by asking Mathematica to evaluate the abstract expression $$\sum\limits_{k = 0}^\infty {\left({- 4} \right)^{- k} \left({\frac{{a_1}}{{4k + 1}} + \frac{{a_2}}{{4k + 2}} + \frac{{a_3}}{{4k + 3}} + \frac{{a_4}}{{4...

For functions of one variable, a unique critical point that is a local maximum is necessarily a global maximum. But this is false in higher dimensions. An example is the graph shown, of the function 3 x e y – x 3 – e 3y : there is only one critical point and it is a local maximum but not a global one.

A variation of the Reuleaux triangle was used in 1939 to describe a device that can drill a perfect square hole. The image shows how, when the outer shape rotates inside a square, the triangular cutting tool traces out an exact square. One can use this to build a device that transforms standard circular motion into motion that drives the Reuleaux r...

The surface is a graph of the reciprocal of the absolute value of the Riemann zeta function ζ (s). The spikes correspond to the zeros on the critical line ½ + iy. Recall that the global behavior of π(x), the prime distribution function, is well approximated by Riemann’s smooth function R(x) (discussed in Chapter 2). More delicate information about...

Using the data in a contour plot, one can devise an algorithm that very efficiently finds all the solutions to a pair of equations f(x, y) = 0, g(x, y) = 0. The example shown has 67 such solutions in the given rectangle.

The top image is the machine-precision trajectory of 0.1 under the quadratic map 4x (1 − x). It suffers from roundoff error and the terms beyond the 60th are not the same as the values in the true trajectory of \(\frac{1}{{10}}\) shown just below it. But it turns out that the noisy trajectory can be shadowed, meaning that there is a value near \(\f...

The image shown is a view of the torus knot known as 83; data for this image, and other aspects of the knot, are available through the KnotData command. Mathematica includes many data sets containing useful and easy-to-use information from physics, chemistry, astronomy, finance, geography, mathematics, and other areas.

VisualDSolve is a Mathematica e-book and accompanying package showing how Mathematica's visualization tools can be used to enhance the viewing of solutions to differential equations. The book contains 18 chapters; the first 5 serve as a manual to the many functions of the package, and the other 13 are comprehensive examples. The last update from 20...

A technique discovered in 1939 can be used to build a device that is driven by standard circular motion (as in a drill press) and drills exact square holes. This device is quite different from the classic design by Watts, which uses a Reuleaux triangle and drills a hole that is almost, but not exactly, square. We describe the device in detail, deri...

Kempe's 1869 proof was flawed, but his ideas can be used to four-color complicated maps.

We discuss the proof of the theorem of D. W. Henderson [Am. Math. Mon. 70, No. 4, 424–426 (1963)] which states that symmetric n-Venn diagrams only exist when n is prime, pointing out what appears to be the omission in the literature of part of the argument, and providing a proof of the missing step.

The impact of computing on mathematics in the last 20 years has been broad and deep, on both the theoretical and applied sides. There are lots of software packages available and for just about any computational task, there is a tool that can handle it. But in terms of coverage of the diverse fields of mathematics, nothing comes close to the newest...

The Frobenius number g(A) of a set A = (a1, a2,..., an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in ave...

Generalizing the familiar pictures of two or three intersecting circles, a Venn diagram is a collection of simple closed curves that intersect in only finitely many points and such that the intersection of interiors of any subset of the curves is nonempty and connected. If there are n curves, and the diagram has n-fold rotational symmetry, n must b...

Might there be a chance of proving in a simple way thatx/π(x) is asymptotic to an increasing function, thus getting another proof of PNT? This is probably wishful thinking. However, there is a natural candidate for the increasing function. LetL(x) be the upper convex hull of the full graph ofxπ(x) (precise definition to follow). The piecewise linea...

Dieses Buch führt seine Leser auf einen packenden Streifzug durch die wichtigsten und leistungsfähigsten Bereiche zeitgenössischer numerischer Mathematik. Die Route orientiert sich an den 10 Wettbewerbsaufgaben der "SIAM 10 x 10-Digit Challenge", mit der Nick Trefethen aus Oxford im Frühjahr 2002 seine Kollegen und ihre Doktoranden weltweit herausf...

The Frobenius problem, also known as the postage-stamp problem or the money- changing problem, is an integer programming problem that seeks nonnegative inte- ger solutions to x1a1 + + xnan = M ,w hereai and M are positive integers. In particular, the Frobenius number f(A), where A =faig ,i s the largestM so that this equation fails to have a soluti...

Foreword Preface The Story 1. A Twisted Tail 2. Reliability amid Chaos 3. How Far Away Is Infinity? 4. Think Globally Act Locally 5. A Complex Optimization 6. Biasing for a Fair Return 7. Too Large to Be Easy Too Small to Be Hard 8. In the Moment of Heat 9. Gradus ad Parnassum 10. Hitting the Ends Appendix A. Convergence Acceleration Appendix B. Ex...

At the 13 th International Snow Sculpting Championship in Breckenridge, CO, Jan 28. – Feb.1, 2003, Team Minnesota, USA, consisting of the above five authors, carved a 12-foot tall, twisted Scherk-Collins Toroid celebrating a 7-year collaboration between Brent Collins and Carlo Séquin. This paper describes the concept, the detailed design, and the i...

ring algorithm for maps. But it is reasonable to take theapproach of first turning the map into a graph, and then use a coloring algorithm forgraphs.2. The Kempe Four-Color AlgorithmIn 1879 Kempe gave an explicit method of 4-coloring planar maps, which I summarizehere from the point of view of planar graphs. Assume a planar graph G is given, withve...