I went Christmas shopping yesterday, to buy a book from the local bookshop that costs £7. I didn't have any money, so I borrowed £5 from Marianne and £5 from Owen. I bought the book and got £3 change. I gave £1 back to each of my kind colleagues and I keep the remaining £1. I now owe each of them £4 and I have £1, giving £9 in total. But I borrowed £10. Where's the missing pound?
There it is!
The answer is that the £10 are a red herring. There's no reason why the money I owe after the whole transaction and the money I still have should add up to £10. Rather, the money I owe minus the change I got should come to the price of the book, that is £7. Giving a pound back to each of Marianne and Owen just re-distributes the amounts. The money I still owe is reduced to £8 and the money I still have to £1. Rather than having £10-£3=£7, we now have £8-£1=£7. Mystery solved!
An infinite set is called countable if you can count it. In other words, it's called countable if you can put its members into one-to-one correspondence with the natural numbers 1, 2, 3, ... . For example, a bag with infinitely many apples would be a countable infinity because (given an infinite amount of time) you can label the apples 1, 2, 3, ... and so on.
Two countably infinite sets A and B are considered to have the same "size" (or cardinality) because you can pair each element in A with one and only one element in B so that no elements in either set are left over. This idea seems to make sense, but it has some funny consequences. For example, the even numbers are a countable infinity because you can link the number 2 to the number 1, the number 4 to 2, the number 6 to 3 and so on. So if you consider the totality of even numbers (not just a finite collection) then there are just as many of them as natural numbers, even though intuitively you'd think there should only be half as many.
Something similar goes for the rational numbers (all the numbers you can write as fractions). You can list them as follows: first write down all the fractions whose denominator and numerator add up to 2, then list all the ones where the sum comes to 3, then 4, etc. This is an unfailing recipe to list all the rationals, and once they are listed you can label them by the natural numbers 1, 2, 3, ... . So there are just as many rationals as natural numbers, which again seems a bit odd because you'd think that there should be a lot more of them.
It was Galileo who first noticed these funny results and they put him off thinking about infinity. Later on the mathematician Georg Cantor revisited the idea. In fact, Cantor came up with a whole hierarchy of infinities, one "bigger" than the other, of which the countable infinity is the smallest. His ideas were controversial at first, but have now become an accepted part of pure mathematics.
We've all got favourite numbers and some of us even go as far as thinking some numbers lucky or unlucky. Seven is probably one that crops up quite a lot in this context, so it's nice to know that it really is a lucky number. No, seriously, that's true! There is a mathematical recipe for finding lucky numbers, and 7 is one of them.
The recipe goes like this. Write down all the positive whole numbers starting from 1. First, you eliminate all even numbers, leaving only the odds:
1, 3, 5, 7, 9, 11, 13, 15, ... .
The next number on the list after 1 is 3, so you now go and eliminate every third number, leaving:
1, 3, 7, 9, 13, 15, ... .
The next surviving number after 3 is 7, so you eliminate every 7th number in the list of survivors. And so on. When you've finished this process the sequence of surviving numbers starts:
These surviving numbers are called lucky numbers. They are similar to the prime numbers in some ways. There are infinitely many of them, and they are the result of a sieving process (successively eliminating numbers) similar to the sieve of Eratosthenes which sieves out primes. When you try and approximate the distribution of the lucky numbers among the other numbers as you move up the number line, the result is akin to the prime number theorem. The lucky numbers even have their own version of the Goldbach conjecture, which states that any odd number greater than 4 can be written as the sum of two primes. One thing that isn't known, though, if whether there are infinitely lucky numbers that are also prime, or only finitely many.
All this doesn't make 7 lucky in the everyday sense of the word — but it makes us feel very lucky to have so many numbers to play with!
Now here's an important mathematical question: how likely are you to
win the lottery?
1 in 14 million.
In the UK lottery you have to choose 6 numbers out of 49, and for a chance
at the jackpot you need all of your 6 numbers to come up in the main draw.
So the question is really how many possible combinations of 6 numbers can
be drawn out of 49? There are 49 possibilities for the first number, 48 for
the second, and so on, to 44 possibilities for the sixth number, so there
are 49x48x47x46x45x44=10,068,347,520 ways of choosing those six numbers... in
But we don't care which order our numbers are picked, and the
number of different ways of ordering 6 numbers are 6x5x4x3x2x1=6! =720: there are six possibilities for the number that comes first in the list, 5 for the number that comes second, and so on.
Therefore our six numbers are one of 49x48x47x46x45x44/6! = 13,983,816 so we
have about a one in 14 million chance of hitting the jackpot. Hmmm...
But on a brighter note, we have just discovered a very useful mathematical fact: the number of combinations of size (sets of objects in which order doesn’t matter) from a larger set of size is:
This sort of argument lies at the heart of combinatorics, the
mathematics of counting. It might not help you win the lottery, but it
might keep you healthy. It is used to understand how viruses such as
influenza reproduce and mutate, by assessing the chances of creating
viable viruses from random recombination of genetic segments.
You can read more on the uses of combinatorics, including in money matters (lottery), love
(well, kissing frogs) and fun (juggling and Rubik's cubes):
Triangles do it, squares do it, even hexagons do it — but pentagons don't. They just won't fit together to tile your bathroom wall. That's the reason why you'll find it very difficult to find pentagonal tiles in any hardware shop.
It's actually really easy to see why pentagons won't tile the plane. We are talking regular pentagons here, shapes that have five sides of equal length and angles between them. In a regular pentagon the internal angle between two sides is 108°. In a regular tiling, adjacent tiles share whole edges, rather than just parts of edges, so corners of tiles meet corners of other tiles. To fit a number of tiles around a corner point, their internal angles must add up to 360°, since that's a full turn. If you try to fit three pentagons, you only get 3 x 108° = 324°, so there is a gap. If you fit four pentagons, you get 4 x 108° = 432°, so two of them overlap. The same isn't true for equilateral triangles, squares, or regular hexagons. Here the internal angles are 60°, 90° and 120° respectively, so you can fit six triangles, four squares and three hexagons around a corner point. (You can try and work out for yourself if any other regular polygons can give you a regular tiling.)
Three pentagons arranged around a point leave a gap, and four overlap. Image: Craig Kaplan.
If pentagons don't work can we perhaps use other shapes with five-fold symmetry to tile the plane? Find out more in The trouble with five.
Remember maps? In the old days, before smart phones and satnavs, we used to use them to get around. They posed quite a few problems, including folding them and understanding them. To help with the latter, map makers often use different colours to colour adjacent regions, be they countries, counties, or parts of a city. And pretty much since the beginning of map making, they have known that they will never need more than four colours to colour a whole map. It's something that's so blindingly obvious once you start playing with a few maps, it should be easy to prove mathematically: when colouring a map drawn on a flat piece of paper, four colours will always suffice.
Mathematicians first set themselves the task back in 1852, and it proved a massive headache. It took 24 years for the first proof to be announced, only to be found faulty another eleven years later. Then it took another 86 years, until 1976, until the mathematicians Kenneth Appel and Wolfgang Haken came up with a second proof. They first supposed that there are maps that need five colours, from which you can then choose one that has the smallest number of countries. They then showed that such a minimal map
must contain one of 1,936 possible configurations; and they also proved that every one of these possible configurations can be reduced into a smaller configuration which also needs five colours. This is a contradiction because they assumed that they had already started with the smallest five-colour map. When you find a contradiction you can deduce that your initial assumption, that there are maps that need five colours, is false.
This sounds good, but there was a hitch. The part of their proof which showed that these 1,936 configurations could be reduced was done by a computer, which simply ploughed through every configuration and checked it. No human being could in their lifetime ever actually read the entire proof to check that it was correct. This caused an outcry: if nobody can check it, how can we ever know it doesn't contain a mistake? Can mathematics ever be done by computers?
The proof has since been improved and verified independently (by computers), so most people are quite happy to consider the result proven. The question of whether computer proofs can ever be considered sound, however, is still a hot one.