Ten of the greatest: Maths puzzles



By CLIFF PICKOVER, Author of The Math Book



From Archimedes' Stomachion to the Tower of Hanoi to the rope around the earth puzzle, here Cliff Pickover chooses his favourite mathematical conundrums





1. ARCHIMEDES' STOMACHION, c250 BC



In 1941, mathematician GH Hardy wrote, 'Archimedes will be remembered when (playwright) Aeschylus is forgotten, because languages die and mathematical ideas do not.' Indeed, the ancient Greek geometer is often regarded as the greatest scientist of antiquity.



In 2002, maths historian Reviel Netz gained a new insight into a treatise by Archimedes concerning a puzzle called the Stomachion. Examining an ancient parchment, he discovered the puzzle involved combinatorics - a field of maths dealing with the number of ways a given problem can be solved.



The goal of the Stomachion is to determine in how many ways 14 pieces can be put together to make a square. In 2003, mathematicians determined that the number is 17,152.





2. WHEAT ON A CHESSBOARD, 1256



The problem of Sissa's Chessboard, discussed by the Arabic scholar Ibn Khallikan in 1256, has been used for centuries to demonstrate the nature of geometric growth, and is one of the earliest puzzles involving chess. According to legend, King Shirham agreed to give a reward consisting of a grain of wheat on the first square of a chessboard, two grains of wheat on the second square, four grains on the third, and so on for the 64 squares. However, he didn't realise how many grains would be awarded. One way to determine the total is to compute the sum of the first 64 terms of a geometrical progression, 1 + 2 + 2<2> + ... + 2<63>, or 2<64> - 1, which is 18,446,744,073,709,551,615 grains of wheat. This would fill a train reaching 1,000 times around the Earth.







3. TOWER OF HANOI, 1883



Invented by French mathematician Edouard Lucas in 1883, the Tower of Hanoi is a puzzle featuring several discs that slide onto any of three pegs.



The discs are initially stacked on one peg in order of size, with the smallest at the top. The goal is to move the entire starting stack to another peg.



You can only move one disc at a time, removing the top disc in any stack and placing it at the top of another.



A disc cannot be placed on top of a smaller disc. The smallest number of moves turns out to be 2<n> - 1, where n is the number of discs. This means that if 64 discs were used and moved at a rate of one per second, finishing the puzzle would take roughly 585 billion years.











4. ROPE AROUND THE EARTH PUZZLE, 1702

This gem from 1702 shows how simple intuition may fail us. Imagine you're given a rope that tightly encircles the equator of a basketball. How much longer would you have to make it for it to be one foot from the surface at all points?

Next, imagine we have the rope around the equator of an Earth-sized sphere - making it around 25,000 miles long. How much longer would you now have to make it for it to be one foot off the ground all the way around the equator?



The surprising answer is 2pi (or approximately 6.28) feet for both the basketball and the Earth. If r is the radius of the Earth, and 1 + r is the radius in feet of the enlarged circle, we can compare the rope circumference before (2pir) and after (2pi(1 + r)).







5. KONIGSBERG BRIDGES, 1736

Graph theory is an area of mathematics that concerns how objects are connected, and often represents problems as dots connected by lines. One of the oldest problems in graph theory involves the bridges of Königsberg in Prussia (now Kaliningrad), linking both sides of a river and two islands. In the early 1700s, people wondered if you could walk across all seven bridges without crossing any bridge more than once, and return to the starting location. In 1736, Swiss mathematician Leonhard Euler proved this was impossible. Today, graph theory is used in the studies of chemical pathways, traffic flow and the social networks of internet users.











Prince Rupert asked: what is the largest wooden cube that can pass through another cube with one-inch sides?

6. PRINCE RUPERT'S PROBLEM, 1816

In the 1600s, Prince Rupert of the Rhine asked a famous geometrical question: what is the largest wooden cube that can pass through another cube with one-inch sides? Perhaps surprisingly, a hole can in fact be made in one of two equal cubes that's sufficiently large for the other cube to slide through - without the cube with the hole falling apart.



Today, we know that a cube with a side length of 1.060660... inches (or smaller) can pass through a cube with one-inch sides. This solution was found by mathematician Pieter Nieuwland and published in 1816. If you hold a cube so that one corner points towards you, you'll see a regular hexagon. The largest square that will squeeze through a cube has a face that can be contained within this hexagon.









7. FIFTEEN PUZZLE, 1874



The Fifteen puzzle caused a real stir in the 19th century. Today, you can purchase a variant of the puzzle with 15 squares (tiles) and one vacant spot in a 4 × 4 frame. At the start, the squares show the numbers 1 through 15 in sequence and then a gap. In a version of the puzzle in Sam Loyd's 1914 Cyclopedia, the starting configuration had the 14 and15 reversed. The goal was to slide the squares up, down, right and left to return them to the correct order. In his Cyclopedia, Loyd claims a prize of $1,000 was offered for a solution; alas, it's impossible to solve the puzzle from this starting position. The original game was developed in 1874 by New York postmaster Noyes Palmer Chapman.





8. THIRTY-SIX OFFICERS PROBLEM, 1779



Consider six army regiments, each consisting of six officers of different ranks. In 1779, Leonhard Euler asked if it was possible to arrange these 36 officers in a 6 × 6 square so that no row or column duplicates a rank or regiment. Euler conjectured that there was no solution, and French mathematician Gaston Tarry proved this in 1901. The problem has led to significant work in combinatorics (see 1). Euler also conjectured that this kind of problem could have no solution for an n × n array if n = 4k + 2, where k is a positive integer. This wasn't settled until 1959, when mathematicians found a solution for a 22 × 22 array.







9. RUBIK'S CUBE, 1974



The Rubik's cube was invented by Hungarian sculptor and professor of architecture Ernö Rubik in 1974.

By 1982, ten million cubes had been sold in Hungary, more than the population of the country. It's estimated that over 350 million have now been sold worldwide. The cube is a 3 × 3 × 3 array of smaller cubes that are coloured in such a way that the six faces of the large cube have six distinct colours.



The 26 external sub-cubes are internally hinged so that these six faces can be rotated. The goal of the puzzle is to return a scrambled cube to a state in which each side has a single colour. In total there are 43,252,003,274,489,856,000 different arrangements of the small cubes. If you had a cube for every one of these 'legal' positions, then you could cover the surface of the Earth (including the oceans) about 250 times.







10. BARBER PARADOX, 1901



In 1901, the British philosopher and mathematician Bertrand Russell uncovered a possible paradox that necessitated a modification to set theory. One version of Russell's Paradox involves a town with one male barber who, every day, shaves every man who doesn't shave himself, and no one else. Does the barber shave himself? The scenario seems to demand that the barber shave himself if and only if he doesn't shave himself!



Russell realised he had to alter set theory so as to avoid such confusion. One way to refute the Barber Paradox might be to simply say that such a barber does not exist. Nevertheless, mathematicians Kurt Gödel and Alan Turing found Russell's work useful when studying various branches of mathematics and computation.









'The Math Book' is published by Sterling, priced at £20.



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