The late Leon Bankoff (he died in 1997) was a Beverly Hills, California, dentist who also was a world expert on plane geometry. (For G. L. Alexanderson’s interview with Bankoff, see [1].) We became good friends. In 1979 he told me about a series of fascinating discoveries he had made about what he called the asymmetric propellor theorem. He intended to discuss them in an article, but never got around to it. This is a summary of what he told me.

The original propeller theorem goes back to at least the early 1930’s and is of unknown origin. It concerns...

In [1] Martin Gardner proved an asymmetric propeller theorem that was originally proposed by Leon Bankoff. He showed that by connecting one vertex from each of three equilateral triangles inscribed in a circle to the center of the circle, one can form a fourth equilateral triangle. This fourth triangle is formed by connecting the remaining vertices with segments as seen in Figure 2.1. Then the midpoints of these segments are connected to form a triangle. This works regardless of the arrangement of the original triangles. We confirmed Gardner’s findings and proceeded to consider his final question: when working with squares,...

If today’s math-and-science-oriented kids could see the resources available to like-minded youths of the 1960s and 70s, they might pity their counterparts back then: no laptop computers, no high-speed internet, no online bookstores. Yet such pity would be wasted on that earlier generation, who could well respond, “Ah, but we had Martin Gardner.”

For a quarter century, Gardner wroteMathematical Games, a wonderful column inScientific American[3]. Each month he treated his readers to a tour of some mathematical recreation that he made tantalizing for people who, like himself, had no advanced training in mathematics. He was at the...

Our dance company, the Dr. Schaffer and Mr. Stern Dance Ensemble, began performing mathematical dance shows in 1990. Scott Kim joined us in 1993, and in 1995 Scott, another Bay area dancer Barbara Susco, and I developed a school-age show entitled “Through the Loop, In Search of the Perfect Square,” which included the performance of many string figures.

Martin Gardner wrote several columns on polyhedra and one on traditional string figures, in addition to his many references to magic tricks and puzzles with string [5, Ch. 1], [6, Ch. 17], [7, Ch. 19], [8, Ch. 10], [6, pp. 74–75]....

More than three hundred years ago, according to the contemporaneous John Wallis [11, pp. 470–471], Prince Rupert* (1619–1682) won a wager that a hole can be cut in one of two equal cubes large enough to permit the second cube to pass through.

Nearly a century later the Dutch scientist Pieter Nieuwland (1764–1794) showed that the largest cube that can be so passed through a cube of side one has side$3\sqrt{2}/4\approx 1.061$. An accessible discussion with enlightening anaglyphs appears in Ehrenfeucht [8]. In 1950, D. J. E. Schreck [10] gave an interesting, historically based survey of...

It’s hard to believe that it was not until 1844 that transcendental numbers were known to exist! But first, a few definitions.

Arational numberis one that can written asa/b, whereaandbare integers. In decimal form, rational numbers either terminate (1/4 = .25) or they have a pattern of consecutive digits that repeat endlessly (1/7 = .142857 142857 142857 …).

Anirrational numberis one that cannot be expressed asa/bwhereaandbare integers. In decimal form, it never ends, and it has no pattern of consecutive digits that keep repeating.

An...

I still recall the thrill and simultaneous disappointment I felt when I first readMathematical Carnival[4] by Martin Gardner. I was thrilled because, as my high school teacher had told me, mathematics was presented there in a playful way that I had never seen before. I was disappointed because Gardner quoted a formula that I thought I had “invented” a few years earlier. I have always had a passion for mental calculation, and the formula (7.1) below appears in Gardner’s chapter on “Lightning Calculators.” It was used by the mathematician A. C. Aitken to square large numbers mentally.

Aitken...

Forms of Nim have been played since antiquity and a complete theory was published as early as 1902 (see [3]). Martin Gardner described the game in one of his earliest columns [7] and returned to it many times over the years ([8]–[16]).

Central to the analysis of Nim is Nim-addition. The Nim-sum is calculated by writing the terms in base 2 and adding the columns mod 2, withno carries. A Nim position is a winning position if and only if the Nim-sum of the sizes of the heaps is zero [2], [7].

Is there is a generalization of...

Alice, the March Hare, the Hatter, and the Dormouse, were standing by the Mad Tea Party (MTP) ride (see opposite page). “We’ll start with the same number of teacups as people,” said the Hatter, bossy as usual. “The teacups are arranged in a circle, and each person sits in his or her own teacup.”

“Won’t that be lonely?” objected Alice.

“Don’t worry,” replied the Hatter, “At the end of this ride, everyone stands up, one of the teacups is removed, then everyone finds a new place to sit. Every teacup has to be occupied by at least one person, so...

Way back in 1852, Francis Guthrie conjectured that every map drawn on the plane can be colored so that regions sharing a border have different colors—and only four colors are necessary. This became known as the Four Color Conjecture. In 1880, Tait [16] proved that the Four Color Conjecture is equivalent to a problem of edge coloring graphs. This is where our story begins, because the study of snarks grew from exactly this edge-coloring problem. To avoid confusion, we note that there is no relationship between the English word ‘snark’ (or its derivativessnarky, snarkiness, etc.) and the mathematical...

Suppose that Alice wants to color a planar map using four colors in aproperway, that is, so that any two adjacent regions get different colors. Despite the fact that she knows for certain that it is eventually possible, she may fail in her first attempts. Indeed, there are usually many proper partial colorings not extendable to proper colorings of the whole map. Thus, if she is unlucky, she may accidentally create such abadpartial coloring.

Now suppose that Alice asks Bob to help her in this task. They color the regions of a map alternately, with Alice...

It has been said that a mathematician can be content with only paper and pencil. In fact, there are times when one doesn’t even need the pencil. From a simple strip of paper it is possible to make a surprisingly interesting geometric object, aflexagon. The flexagon can credit its creation to the difference in size between English-ruled paper and American binders. The father of the flexagon, Arthur Stone, was an English graduate student studying at Princeton University in 1939. To accommodate his smaller binder, Stone removed strips of paper from his notebook sheet. Not being wasteful, he creased these...

Using only the pinch flex, the flex described by Martin Gardner in [4], the six triangles of each hexagonal face of a hexaflexagon stay together. If the faces are colored, the face facing up is always monochrome. To scramble the triangles and mix the colors, you need other flexes. In this paper, we describe the V-flex. With the V-flex, faces become multi-colored when flexed. It takes persistence to master, but the V-flex is worth it. A hexahexaflexagon has only 9 mathematical faces under the pinch flex; with the V-flex it has 3,420.

We conjecture that many people performed a V-flex...

Hexaflexagons, the subject of Martin Gardner’s column [5], were the first family of flexagons to be discovered. They are, however, onlyoneexample of anedge flexagon, a folded band of hinged, usually identical, convex polygons (calledleaves). The strip of polygons used to construct a flexagon is called anet. In amain positionan edge flexagon has the appearance of a ring of hinged polygons (see Figure 14.1). The rings are not necessarily flat and what look like single polygons are folded piles of one or more leaves, calledpats. By definition, edge flexagons can be flexed to...

Flexagons are remarkable objects obtained by suitably folding and glueing a strip of paper. Several websites give detailed instructions on how to construct various kinds. After construction, the flexagon lies flat with a number of faces, delineated by creases in the paper, visible on top and bottom. The name derives from the fact that they can be pinched, or flexed, in various ways into a 3-dimensional shape and then flattened again to reveal different faces. First discovered by Arthur Stone at Princeton in 1939, they were popularized by Martin Gardner [2] in his 1956 debut column inScientific Americanand...

The well known Catalan numbersC_nare named after Belgian mathematician Eugene Charles Catalan (1814–1894), who found them in his investigation of well-formed sequences of left and right parentheses. As Martin Gardner (1914–2010) wrote inScientific American[2], they have the propensity to “pop up in numerous and quite unexpected places.” They occur, for example, in the study of triangulations of convex polygons, planted trivalent binary trees, and the moves of a rook on a chessboard [1, 2, 3, 4, 6].

The Catalan numbersC_nare often defined by the explicit formula${{C}_{n}}=\frac{1}{n+1}\left( \begin{matrix} 2n\\ n\\ \end{matrix} \right)$, wheren≥ 0...

Suppose a standard chessboard is ‘mutilated’ by the removal of two diagonally opposite corner cells. Can the remaining 62 squares be tiled with 31 dominos? The answer is ‘no’ because the removed squares are thesamecolor. Say the color is white. The remaining 62 squares will have an excess of two black cells. Each domino covers one black and one white cell. After 30 are placed, two black cells will remain uncovered. They cannot be adjacent, therefore they can’t be covered by a domino. This famous puzzle, solved by a simple parity check, is a simple example of a...

Those of us fortunate to have been born before Martin Gardner’s retirement can relate to the excitement of eagerly awaiting his monthly columnMathematical Gamesin the magazineScientific American. Particularly popular were those containing a selection of short puzzles. Here is one from the very last column of this kind, for April, 1981.

The patchwork quilt in Figure 18.1 was originally made up of 108 unit squares. Part of the quilt’s center became worn, making it necessary to remove 8 squares as indicated. Cut the quilt along the lines into just two parts that can be sewn together to...

This research was inspired by two lovely pieces of mathematics. The first is the discovery by William T. Tutte, A. H. Stone, R. L. Brooks, and C. A. B. Smith of squares with integral sides that can be tiled by smaller squares with integral sides, no two alike. Tutte tells the story in “Squaring the Square,” a beautifully written article that conveys vividly the excitement of mathematical research [9]. It became widely-read in 1958 when it was reprinted in Martin Gardner’sMathematical Gamescolumn inScientific American. It undoubtedly played a role in inspiring many to become mathematicians.

The second...

InMathematical Magic Show[4], Martin Gardner looked at the classic problem of the knight’s tour on a chess board, paying particular attention to tours in which the numbers in each row and column added to 260. At that time, the question of whether the numbers in each long diagonal could also be made to add to 260 was still open. Thanks to advances in computer power since Martin wrote, this question has now been decided, and perhaps readers will be interested in the complete statement that it is now possible to make.

In knight’s tour literature, the termmagic...

Combinatorial problems involving magic squares, stars, and other geometrical structures often can be solved by brute force computer programs that simply explore all possible permutations of numbers. When the number of permutations is too large for a feasible running time, an algorithm can frequently be reduced to manageable time by finding ingenious shortcuts. Such planning makes computer solving less trivial and much more interesting.

A superb example of such planning was described in a little-known short article in theMathematical Gazette(vol. 75, June 1991, pp. 140–142). The authors, Brian Bolt, Roger Eggleton, and Joe Gilks, posed for the...

Problem 394 in Henry Dudeney’s536 Puzzles and Curious problems[1] poses this puzzle: Put the numbers from 1 to 12 in the circles on the left in Figure 22.1 so that the sum of the four numbers on each line is 26. One solution is shown on the right.

The solution pictured in Figure 22.1 is listed as solution number 16 in Table 22.1 at the end. Table 22.1 contains 20 solutions to the puzzle. The letters in Figure 22.1 on the left are used as positions for the solutions in Table 22.1.

There are many puzzles referred to...

One of the first of Martin Gardner’sMathematical Gamescolumns I ever read was “Euler’s Spoilers,” in November 1959 [1], and after all these years I think it is still my favorite (maybe this is just because I love the way he rhymed ‘Euler’ and ‘spoiler’). It dealt with what we now call Latin squares,n×narrays usingnsymbols such that each symbol appears exactly once in each row and column.

Latin squares are extremely useful in the design of statistical experiments, but they are perhaps better known now for their appearance in recreational puzzles such as...

Martin Gardner was fond of mathematically-based magic tricks using simple apparatus—in particular, pennies. Chapter 2 of hisMathematical Carnival[2, pages 12–26] is entirely about penny puzzles. About five years later, Gardner and Karl Fulves invented a delightfully simple trick with three pennies (see Demaine [1]). Its explanation involves an important mathematical idea called a state diagram, useful in many combinatorial problems.

A related trick, usually described using cups instead of coins, has the same state diagram with ‘heads’ and ‘tails’ replaced by ‘up’ and ‘down.’ This article explores the relations between the two tricks, develops the deeper...

“Oh, you’re a mathematician! Let me show you something interesting.”

But I’m trying to work on my talk, I thought. As the train sped along, the man sitting across from me looked eager.OK, let’s get this over with.

“Here are fifteen playing cards. Arrange them into piles; as many piles as you like, each with as many cards as you like.”

I made five piles with heights 3, 1, 4, 1, 6. Theπreference went unnoticed.

“Now take one card from each pile to make a new pile.”

The operation left me with piles of 3 − 1...

An expedition into Congo uncovered a treasure chest in the shape of a regular octagon. At each corner was a bongo drum. A scroll attached to the chest, written in French, explained that there was a genie inside each bongo drum. A genie is either standing upright or doing a handstand. One may strike a number of bongo drums at the same time. When a bongo drum is struck, the genie inside will change its posture from right side up to upside down, or vice versa. The treasure chest will open if and only if all genies are right side...

In his classic collectionCyclopedia of Puzzles, published in 1914, Sam Loyd has two versions of the Courier Problem ([2, p. 315]):

For the reason that many communications are being received relating to a very ancient problem, the authorship of which has been incorrectly accredited to me, occasion is taken to present the original version which has led to considerable discussion. It has been reproduced, in many forms, generally accompanied by an absurd statement regarding the impossibility of solving it, which produced letters of inquiry as well as correct answers from some, who, under the misapprehension of having mastered a...

Martin Gardner created no less than a revolution in the popularization of mathematics through his many books, and, in particular, hisScientific Americancolumns. Some of his most important columns, which spurred not only popular interest but also a wealth of innovative research and even practical applications, were his columns describing John Conway’s “Game of Life” [5, 6]. Gardner himself said [7],

Probably my most famous column was the one in which I introduced Conway’s game of Life. Conway had no idea, when he showed it to me, that it was going to take off the way it did. He...

In 1907, the Dutch mathematician, Willem Abraham Wythoff [13] invented this game, later vividly explained by Martin Gardner in [7].

WYTHOFF is played on a pair of nonnegative integers, (M,N). A move consists of either (i) subtracting any positive integer from preciselyoneofMorNsuch that the result remains nonnegative, or (ii) subtracting thesamepositive integer from bothMandNsuch that the results remain nonnegative. The first player unable to move loses.

Given the position (3, 3), say, the next player wins in a single move: (3, 3) → (0, 0). The position...

Because playing cards have values 1 through 13 (jacks 11, queens 12, kings 13), come in two colors, four suits, and have fronts and backs, they provide wonderfully convenient models for hundreds of unusual mathematical problems involving number theory and combinatorics. What follows is a choice selection of little known examples.

One of the most surprising of card theorems is known as the Gilbreath principle after magician Norman Gilbreath who first discovered it. Arrange a deck so the colors alternate. Cut it so the bottomcards of each half are different colors, and then riffle shuffle the halves together. Take cards...

The Ashland University student chapter of the MAA holds biweekly meetings, typically consisting of a short business meeting to plan activities that the group is sponsoring followed by a social time when the group plays mathematical games and munches on brownies. As a new feature in Fall 2002, I told the students that I would perform a new mathematical card trick at each of these meetings. My source for most of these card tricks is the delightful book [2]. One of the finest tricks described there is due to an amateur New York magician named Henry Christ. A spectator shuffles...

In its classical form, the Monty Hall Problem (MHP) is the following:

Version 1. (Classic Monty) You are a player on a game show and are shown three identical doors. Behind one is a car, behind the other two are goats. Monty Hall, the host of the show, asks you to choose one of the doors. You do so, but you do not open your chosen door. Monty, who knows where the car is, now opens one of the doors. He chooses his door in accordance with the following rules:

1. Monty always opens a door that conceals a goat.

2. Monty...

Searching for a job is always stressful and, with unemployment rates at their highest levels in years, never more so than now. Applicants can and should use every advantage at their disposal to obtain a job which is rewarding, financially and otherwise. While this author believes a math major gives applicants many advantages as they search for their dream job, one often overlooked is the ability to strategize and schedule their interviews to maximize the chance of landing that job.

The secretary problem helps an employer find the optimal candidate for a job out of a large pool of applicants....

Nontransitive dice, discovered by Bradley Efron, but introduced to the public by Martin Gardner [2], are sets of three dice which if rolled in pairs, dieAmost frequently beats dieB,BbeatsC, andCbeatsA. This is a statistical analogue of the gameRock, paper, scissors(also calledrochambeau, roshambo,andjan-ken-pon), in which each element or action beats another and is beaten by yet another. In such a set there is no best or dominant element. Of course the scalar mean values of the dice cannot exhibit nontransitivity; it is only in the statistical case...

Martin Gardner was amazingly accurate and reliable. That he made a mistake is simply testimonial to the difficulty of this particular problem, which appeared in 1959 and was republished in [3]:

Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?

Mr. Jones has two children. The older child is a girl. What is the probability that both children are girls?

Mr. Jones has failed to stir any controversy, so we ignore him and his two children [5]. Instead, we concentrate on Mr. Smith. Here is the solution...

Luther Washington was a friendly, shy, intelligent boy, the oldest of five children who lived with their parents in Butterfield, Kansas. He was one of sixteen African Americans who attended Butterfield Central High. His father owned a small grocery store in the town’s black district. His mother cooked dinners for one of the town’s bankers.

For some reason, which his parents never understood, Luther was fascinated by numbers. One day, when he was ten, he surprised his father by saying: “Dad, I’ve discovered an easy way to tell if a big number can be divided by four or eight and...

The following piece is one of a series of results that I called “modular miracles” in a 2001 article in theMonthly. It depends on an earlier miracle, which I must therefore briefly explain. An 1857 theorem of Kronecker [3] relates the famousmodular function, j,to unique prime factorization in the field$\mathbb{Q}(\sqrt{-D})$, whereDis a positive integer. Kronecker’s theorem states that unique prime factorization holds for the integers τ in$\mathbb{Q}(\sqrt{-D})$only whenj(τ) is an ordinary integer for each such τ.

Take, for example, theGaussian integers,${a+b}{\sqrt{-1}}$for ordinary integersaandb. These...

Over the past five centuries, a great deal of nonsense has been written about the golden ratio,$\Phi =\frac{1+\sqrt{5}}{2}$, its geometry, and the Fibonacci sequence. Many authors make claims that these mathematical entities are ubiquitous in nature, art, architecture, and anatomy. Gardner [4] has shown that the admiration for this number seems to have been raised to cult status. Fortunately, however, there have been some recent papers, including Fischler [2] in 1981, Markowsky [7] in 1992, Steinbach [9] in 1997, and Fowler [3] in 1982, that are beginning to set the record straight. For example, Markowsky, in his brilliant paper...

In 1975, Martin Gardner wrote in hisMathematical Gamescolumn inScientific Americanabout Roger Penrose’s discovery of a set of tiles that do not tile the plane periodically, but do tile it nonperiodically. Gardner did not show pictures because Penrose was waiting for a patent. In response to that column, Gardner received a letter from Robert Ammann, who wrote, “I am also interested in nonperiodic tiling, and have discovered both a set of two polygons which tile the plane only nonperiodically and a set of four solids which fill space only nonperiodically.” Ammann included pictures: he had independently discovered...

This is the only novel I know of that is saturated with mathematics. Alice Butler, the 29-year-old narrator of the story, works for the world’s third-largest toy company, PopCo. Her job is to create ideas for toys and games, especially products that will appeal to teen-age girls.

Orphaned when young, Alice is raised by loving and loved grandparents, both of whom are first-rate mathematicians. Her grandfather writes a monthly column on recreational math, said to be similar to the one I wrote forScientific American. He is obsessed with trying to decode the notorious Voynich manuscript, now widely believed to...

Several years ago I was a graduate student at the University of Chicago. I was working on my doctorate in physics, about possible ways to test superstring theory, when my brother in Tulsa died suddenly from a heart attack. Both parents had earlier passed away. After the funeral I drove past my past, marveling at the enormous changes that had taken place since I grew up there. I drove by the red brick building, now an enormous warehouse, that had once been Tulsa Central High. My grades in history, Latin, and English lit were low, but I was good in...