The number my5teries solutions

Puzzle 1: the curious incident of the never-ending numbers There are 78 presents in total. Mathematicians are lazy at heart so we like quick ways to calculate things. So here is a way to make a formula for the presents using a bit of geometry. Stack the presents in a triangle, as below. At the top is one pear tree (P). The second layer contains two turtle doves (T). The third lay contains three French hens (F). The last layer of the triangle (not pictured) would contain 12 drummers drumming along the bottom. Now take an another copy of this triangle, turn it upside down and place it on top of the first triangle. There are 12 x 13 boxes in total in this rectangle of presents you’ve built. But this rectangle has twice as many presents as we are trying to calculate. So divide this number by 2 to get 12x13/2=78 presents.

The geometric trick explained with the presents received on the third day of Christmas:

The million dollar question: The Riemann Hypothesis. This prize is associated with another sequence of numbers which doesn’t seem to have any formula to help us explain them: the primes. A prime number is a number only divisible by itself and 1, like 17 and 23. The sequence starts with 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... The Ancient Greeks proved that the primes continue for ever.

Can you spot the pattern in the primes?

Puzzle 2: the story of the elusive shapes The astronaut lives on a doughnut or bagel or what mathematicians call a torus. Since the astronaut flies off the bottom and comes on at the top, we can join these two bits of the universe up to make a cylinder. But when he flies off the left he rejoins the world on the right hand side of the screen. So the two ends of the cylinder need to be joined to make a bagel shape. The letters and numbers (A, B, 1, 2) show how that universe needs to be joined up.

The million dollar question: The Poincaré Conjecture. What are the possible mathematical shapes that our three-dimensional universe could be? Is it a hyperbagel or can it be something more exotic? It may be that this million will soon be claimed. A Russian mathemati

cian, Grigori Perelman, grabbed the headlines this summer with the news that he might have cracked the search for the elusive shapes.

Puzzle 3: the secret of the winning streak 238 metres is the shortest path round the penguins. But don’t despair if you couldn’t find a clever way to the solution. Mathematicians can only solve this by trial and error.

The million dollar question: P versus NP. This is an example of NP-complete problem or “a needle in a haystack” problem. Called the Travelling Salesman Problem, the only way to guarantee finding the shortest path seems to be by trying them all out. But is this the only way or is there a clever mathematical trick that we’re missing to find the solution? Sorting out the complexity of the Travelling Salesman Problem would win you a million dollars.

Puzzle 4: the case of the uncrackable code The new album is called XMAS. Coldplay’s Album cover is based on the Baudot code where each letter of the alphabet is changed into a string of five 0s and 1s. So the letter X in Baudot is 10111. Instead of 0s and 1s Coldplay use coloured blocks. If you look at the first column of colours on the cover, you see a black and grey block representing a 1, followed by a gap representing a zero then three more blocks of colour giving three 1s. The first letter in Coldplay’s title is the letter X.

The Frenchman Emile Baudot devised his code in 1870 and it was ideal for transmitting messages across the growing network of telegraph wires.

The million dollar question: The Birch and Swinnerton-Dyer Conjecture. Prime numbers are key to the codes that protect electronic commerce on the internet. But there are even more sophisticated codes depending on equations called elliptic curves which are used for secure communication across mobile phones. One of the million dollar prizes is for the person who can understand how to solve these equations.

Puzzle 5: the quest to predict the future (i) The population goes from 2 to 3 to 4 and then stabilises at 5 from then on. Even if you start with a different number of reindeer (fewer than 10) the population always stabilises at 5.

(ii) When the reindeer triple, the population goes from 2 to 5 to 8 and then ping-pongs back and forth each year between 5 and 8 reindeer. Start from a different number of reindeer and you get a similar ping-pong effect. For example, starting with 3 leads to reindeer alternating between 6 and 7 each year.

(iii) In the model where reindeer quadruple, the population always dies out! A lesson not to be too promiscuous at Christmas time.

The million dollar question: The above is a simplified version of a model discovered by Bob May which illustrates how chaos theory underlies population growth. Chaos theory is at the heart of many of Nature’s most unpredictable forces. For example, the turbulence behind an airplane wing is described by the Navier–Stokes equations which also demonstrate chaotic behaviour. Solving these equations will win you the last million on offer in this year’s Christmas Lectures.