The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du
player cuts out three Subbuteostyle players and chooses different prime numbers to write on their backs. Use one of the Euclid footballs from Chapter 2 (page 66).
The ball starts with a player from Team 1. The aim is to make it past the three players in the opponent’s team. The opponent chooses the first player to try to tackle Team 1’s player. Roll the dice. The dice has six sides: white 3, white 5 and white 7, and black 3, black 5 and black 7. The dice will tell you to divide your prime and the prime of your opponent’s player by 3, 5 or 7 and then take the remainder. If it is a white 3, 5 or 7, your remainder needs to equal or beat the opposition. If it is black, you need to equal or get less than your opponent.
To score, you must make it past all three players and then go up against a random choice of prime from the opposition. If at any point the opposition beats you, then possession switches to the opposition. The person who has gained possession then uses the player who won to try to make it through the opposition’s three players. If Team 1’s shot at goal is missed then Team 2 takes the ball and gives it to one of their players.
The game can be played either against the clock or first to three goals.
This may sound totally irrational coming from a mathematician, someone who is meant to be a logical analytical thinker. However, I also play in a prime number shirt for my football team, Recreativo Hackney, so I felt some connection with the man in 23. My Sunday League team isn’t quite as big as Real Madrid and we didn’t have a 23 shirt, so I chose 17, a rather nice prime—as we’ll see later. But in our first season together our team didn’t do particularly well. We play in the London Super Sunday League Division 2, and that season we finished rock bottom. Fortunately this is the lowest division in London, so the only way was up.
But how were we to improve our league standing? Maybe Real Madrid were on to something—was there was some psychological advantage to be had from playing in a prime number shirt? Perhaps too many of us were in non-primes, like 8, 10 or 15. The next season I persuaded the team to change our kit, and we all played in prime numbers: 2, 3, 5, 7, … all the way up to 43. It transformed us. We got promoted to Division 1, where we quickly learnt that primes last only for one season. We were relegated back down to Division 2, and are now on the look-out for a new mathematical theory to boost our chances.
Should Real Madrid’s keeper wear the number 1 shirt?
If the key players for Real Madrid wear primes, then what shirt should the keeper wear? Or, put mathematically, is 1 a prime? Well, yes and no. (This is just the sort of maths question everyone loves—both answers are right.) Two hundred years ago, tables of prime numbers included 1 as the first prime. After all, it isn’t divisible, since the only whole number that divides it is itself. But today we say that 1 is not a prime because the most important thing about primes is that they are the building blocks of numbers. If I multiply a number by a prime, I get a new number. Although 1 is not divisible, if I multiply a number by 1 I get the number I started with, and on that basis we exclude 1 from the list of primes, and start at 2.
Clearly Real Madrid weren’t the first to discover the potency of the primes. But which culture got there first—the Ancient Greeks? The Chinese? The Egyptians? It turned out that mathematicians were beaten to the discovery of the primes by a strange little insect.
Why does an American species of cicada like the prime 17?
In the forests of North America there is a species of cicada with a very strange life cycle. For 17 years these cicadas hide underground doing very little except sucking on the roots of the trees. Then in May of the 17th year they emerge at the surface en masse to invade the forest: up to a million of them appearing for each acre.
The cicadas sing away to one another, trying to attract mates. Together they make so much noise that local residents often move out for the duration of this 17-yearly invasion. Bob Dylan was inspired to write his song ‘Day of the Locusts’ when he heard the cacophony of cicadas that emerged in the forests round Princeton when he was collecting an honorary degree from the university in 1970.
After they’ve attracted a mate and become fertilized, the females each lay about 600 eggs above ground. Then, after six weeks of partying, the cicadas all die and the forest goes quiet again for another 17 years. The next generation of eggs hatch in midsummer, and nymphs drop to the forest floor before burrowing through the soil until they find a root to feed from, while they wait another 17 years for the next great cicada party.
It’s an absolutely extraordinary feat of biological engineering that these cicadas can count the passage of 17 years. It’s very rare for any cicada to emerge a year early or a year too late. The annual cycle that most animals and plants work to is controlled by changing temperatures and the seasons. There is nothing that is obviously keeping track of the fact that the Earth has gone round the Sun 17 times and can then trigger the emergence of these cicadas.
For a mathematician, the most curious feature is the choice of number: 17, a prime number. Is it just a coincidence that these cicadas have chosen to spend a prime number of years hiding underground? It seems not. There are other species of cicada that stay underground for 13 years, and a few that prefer to stay there for 7 years. All prime numbers. Rather amazingly, if a 17-year cicada does appear too early, then it isn’t out by 1 year, but generally 4 years, apparently shifting to a 13-year cycle. There really does seem to be something about prime numbers that is helping these various species of cicada. But what is it?
While scientists aren’t too sure, there is a mathematical theory that has emerged to explain the cicadas’ addiction to primes. First, a few facts. A forest has at most one brood of cicada, so the explanation isn’t about sharing resources between different broods. In most years there is somewhere in the United States where a brood of prime number cicadas is emerging. 2009 and 2010 are cicadafree. In contrast, 2011 sees a massive brood of 13-year cicadas appearing in the south-eastern USA. (Incidentally, 2011 is a prime, but I don’t think the cicadas are that clever.)
The best theory to date for the cicadas’ prime number life cycle is the possible existence of a predator that also used to appear periodically in the forest, timing its arrival to coincide with the cicadas’ and then feasting on the newly emerged insects. This is where natural selection kicks in, because cicadas that regulate their lives on a prime number cycle are going to meet predators far less often than non-prime number cicadas will.
FIGURE 1.02 The interaction over 100 years between populations of cicadas with a 7-year life cycle and predators with a 6-year life cycle.
For example, suppose that the predators appear every 6 years. Cicadas that appear every 7 years will coincide with the predators only every 42 years. In contrast, cicadas that appear every 8 years will coincide with the predators every 24 years; cicadas appearing every 9 years will coincide even more frequently: every 18 years.
FIGURE 1.03 The interaction over 100 years between populations of cicadas with a 9-year life cycle and predators with a 6-year life cycle.
Across the forests of North America there seems to have been real competition to find the biggest prime. The cicadas have been so successful that the predators have either starved or moved out, leaving the cicadas with their strange prime number life cycle. But as we shall see, cicadas are not the only ones to have exploited the syncopated rhythm of the primes.
Cicadas v predators
Cut out the predators and the two cicada families. Place predators on the numbers in the six times table. Each player takes a family of cicadas. Take three standard six-sided dice. The roll of the dice will determine how often your family of cicadas appears. For example, if you roll an 8,