Adventures in Recreational Mathematics

The following sections are included:

• Dedication
• Contents of Volume 2
• Preface
• About the Author

This chapter is inspired by Eugene Wigner’s famous essay “The unreasonable effectiveness of mathematics in the physical sciences” [35]. Unlike Wigner, after much thought, we present an explanation of why recreational mathematics is, if not “reasonable”, at least useful. Wigner was concerned with the philosophy of science, but our concerns are pragmatic, not philosophical.

Some problems just call out for attention. Which fits better, a round peg in a square hole or a square peg in a round hole? This can easily be solved once one arrives at the following mathematical formulation of the problem. Which is larger: the ratio of the area of a circle to the area of the circumscribed square or the ratio of the area of a square to the area of the circumscribed circle? One easily finds that the first ratio is π/4 (= 0.785 …) and that the second is 2/π (= 0.636 …). Since the first is larger, we may conclude that a round peg fits better in a square hole than a square peg fits in a round hole…

Most readers will have heard this famous bear hunting problem. The standard answer is “white” because he can travel in the stated way if and only if he starts at the North Pole…

Whether we stopped to think about it or not, we are all familiar with 2 + 2 = 2 · 2 and 1 + 2 + 3 = 1 · 2 · 3 as examples where a sum of positive integers is equal to their product. It is a curious fact and we wonder if such examples are scarce. Andrew Palfreyman [1] discusses such sequences with equal sums and products.¹ As will be seen, there are a number of unsolved questions that arise and these led to some correspondence with Mike Bennett and Andrew Dunn and an article in the Fibonacci Quarterly (1995)…

The following sections are included:

• The Original Puzzle
• Further Problems
• Bibliography

Pascal’s triangle has been a source of problems for centuries. Everyone who has been shown some hidden pattern in that famous array of numbers, wonders if they can find some new pattern? Probably not without some effort as this is a well-plowed field. However, we have found a new problem to explore in this chapter. If you look, you will find that repetitions of numbers are rare and any given entry is repeated very few times. Recall the entries of Pascal’s triangle are the binomial coefficients, so we will just use the term binomial coefficient…

One of the delights of mathematics is discovering different routes to the same result. It displays the nature of unseen connections that some people find as evidence that mathematics is “discovered” rather than “invented”. Many find searching for new proofs of known results simultaneously fun and sublime. Finding ways to prove identities leads to pleasant and surprising insights…

Königsberg, now Kaliningrad, in the Kaliningrad Oblast of the Russian Federation, was the capital of East Prussia and a German city until World War II (Figure 8.1). It is a few miles up the River Pregel [Pregolya in Russian] from the Baltic Sea. The Oblast (=region) is disconnected from the rest of Russia and found between modern Poland, Lithuania and the Baltic Sea…

In 1988, Tim Sole [9] asserted that in the 4, 5, 6 triangle, one angle was double another angle. This is the kind of observation that leads the mathematically minded to ask if there are more examples. This thinking led the Danish mathematician Mogens Larsen to explore this problem. His preliminary results appeared as “The Triangle with One Angle Twice Another”. He showed that this leads to the relation a² = b² + bc among the sides. In particular, he proved ∠A = 2∠B ↔ a² = b² + bc. The preprint led to correspondence. And as often happens correspondence led to generalizations. What we both discovered are the results in this chapter.

All members of London South Bank University have probably noticed that the previous South Bank Polytechnic’s coat of arms featured an unusual geometric pattern of a regular pentagon surrounded by five other regular pentagons (Figure 10.1)…

The Wobbler is a geometric device invented by Frederick Flowerday some years ago. It consists of two interlocked circles. The planes of the circles are orthogonal and the centers of the circles are separated by $\sqrt{2}$ times the common radius. In practice, it is made from two circular discs, each with a radial notch. When the Wobbler is rolled on a plane, it executes a most fascinating wobbly motion. Observation indicates that the center of gravity remains at, or nearly at, constant height. We demonstrate that it does remain at constant height and that this holds if and only if the separation of centers is $\sqrt{2}$ times the radius. We also study the distance between the points of contact and show that this is constant if and only if the separation of centers is equal to the radius. Such a device was invented by Paul Schatz [1]…

If it is not easy to visualize the geometry then consider a huge version, called "Rocking Toy" that was made by the artist Anthea Alley, in Figure 11.1. The holes are for artistic reasons only.

Getting students to carry out calculations can be difficult unless one embeds the calculations in an amusing or interesting setting. This chapter describes several calculations which have intrigued students for centuries and ought to intrigue students for centuries to come. This leads to large numbers and the interest of the problem is to make these numbers comprehensible. The problems include: the chessboard problem; the landowner’s earth and air problem; the value of Manhattan Island; heavy rainfall; permutations of the alphabet and the number of crosswords; grains of sand versus stars in the sky; winning the lottery versus dying; the size of a million pounds. Each of these leads to large numbers and the interest of the problem is to make these numbers comprehensible. In addition, the problems lead to questions of measurement, area, volume, the size and population of the earth, interest rates, etc., and some research may be needed to determine the relevant numbers. Each student or group can be encouraged to produce improved measurements, variations on the method or variations on the problem itself. As a further variation, one can convert into other systems of units — in first year physics, we had to convert the speed of light into furlongs per fortnight.

The Three Rabbits image was first presented as a puzzle in the 19th century. It is a curious blend of a paradox and an optical illusion. The earliest example seen is in The Girl’s Own Book, a popular 19th century American book. The first edition apparently was in 1831 and 42 editions have been traced in the US and Britain [1]. Here is a version from a facsimile of the 1833 edition.