Professor Stewart's Hoard of Mathematical Treasures (39 page)

Unlike Fermat, he was wrong. In 1966 Leon Lander and Thomas Parkin discovered that
This remained the only known example of the failure of Euler’s conjecture until 1988, when Noam Elkies discovered that
In fact, Elkies proved that there are infinitely many cases where three fourth powers add up to a fourth power - but most of them require very big numbers. Roger Frye used a computer to search by trial and error, and found the smallest example:

Suppose we write out all whole numbers in turn, strung together like this:
and so on.

What is the millionth digit?

 

Roger Redbeard, the fiercest pirate in the Kidnibbean Sea, has forgotten a vital piece of information - the address of his bank in the Banana Islands, where he keeps his loot safe from the attentions of the tax authorities. He knows which street it is on, but there are more than thirty banks on Taxhaven Street, all nameless, all looking exactly alike.

All is not lost, however, because he has a map.

Redbeard’s map.

The address of his bank is cunningly concealed in this map: it is the number of distinct ways to trace the word PIRATE, starting at the circle marked P and spelling out the word letter by letter to end at the circle marked E. The address is the number of different ways that this can be achieved, always moving along the lines linking the letters.

What is the address of Redbeard’s Bank?

 

Two trains, the Atchison Flier (A) and the Topeka Bullet (B), are travelling in opposite directions towards each other along the same single-line track. Each consists of one locomotive, at the front, and nine coaches. Both locomotives and all coaches have the same length. The siding can accommodate no more than four coaches or locomotives in total at any one time, while leaving room for trains to pass along the main track.

Can the trains pass each other? If so, how?

 

Answer on page 325.
[Hint: coaches can be decoupled.]

We’re stuck - aren’t we?

The mathematical logician Abraham Fraenkel, who was of German origin, once boarded a bus in Tel Aviv, Israel. The bus was scheduled to depart at 9.00 precisely, but by 9.05 it was still sitting in the bus station.

Aggrieved, Fraenkel waved a timetable at the driver.

‘What are you - a German or a professor?’ the driver enquired.

‘Do you mean the inclusive or, or the exclusive or?’ Fraenkel replied.
³⁸

Abraham Fraenkel.

The list
consists of seven consecutive squares. It has a curious feature: the sum of the decimal digits of each of these numbers is itself a square. For example 1 + 6 + 9 = 16 = 4
²
.

Find another sequence of seven consecutive squares with the same property.

 

In 1900, the German mathematician David Hilbert gave a famous lecture to the International Congress of Mathematicians in Paris, in which he listed 23 of the most important problems in mathematics. He didn’t list Fermat’s Last Theorem, but he mentioned it in the introduction. Here’s a potted description of Hilbert’s problems, and their current status.

1.
Continuum Hypothesis

2.
Logical Consistency of Arithmetic

3.
Equality of Volumes of Tetrahedra

4.
Straight Line as Shortest Distance Between Two Points

5.
Lie Groups Without Assuming Differentiability

6.
Axioms for Physics

7.
Irrational and Transcendental Numbers

8.
Riemann Hypothesis

9.
Laws of Reciprocity in Number Fields

10.
Determine When a Diophantine Equation has Solutions

11.
Quadratic Forms with Algebraic Numbers as Coefficients

12.
Kronecker’s Theorem on Abelian Fields

14.
Finiteness of Complete Systems of Functions

15.
Schubert’s Enumerative Calculus

16.
Topology of Curves and Surfaces

17.
Expressing Definite Forms by Squares

18.
Tiling Space with Polyhedra