From 0 to Infinity in 26 Centuries (20 page)

Most people’s first response to the conundrum goes something along the lines of: there would need to be 366 people in the room to guarantee two of them sharing a birthday, so 183 people
(half of 366) in the room would give a 50% chance.

The correct answer, however, is only 23 people. Although this seems very unlikely, it is in fact true. Why?

Well, if there are two people in the room, Alan and Blaise, there is only one way they can both share a birthday. If a third person, Carl, joins them, then there are three possible matches:

AB BC

AC

If a fourth person, Delia, joins them, the possible matches are:

AB BC CD

AC BD

AD

Now there are six chances for two people in the room to share a
birthday. Each time a new person joins the room another layer is added to the
triangle:

AB BC CD DE

AC BD CE

AD BE

AE

With five people in the room there are ten combinations that could result in a shared birthday. This triangular pattern stops you from having to add further layers because the
triangular numbers
that the pattern produces are well known to mathematicians.

Even more triangles!

Triangular numbers are formed by adding together consecutive whole numbers. Hence, the first triangular number is 1, the second is 1 + 2 = 3, the third is 1 + 2 + 3 = 6 etc.

The formula for the nth triangular number is ½ × n × (n+1), but I can also find the triangular numbers using Pascal’s triangle
(see
here
), by looking down the third diagonal:

The triangular numbers also have a role to play in the carol ‘The Twelve Days of Christmas’. In this song, each day your ‘true love’ gives to you an
ever-increasing quantity of gifts:

As you can see, the number of gifts awarded each day corresponds to the triangular numbers. But what about the running total of gifts? These numbers are the next diagonal of
Pascal’s triangle and are known as the
tetrahedral numbers
, because each time you add another day you are adding another layer to the pyramid:

The formula for these numbers is n × (n+1) × (n+2) ÷ 6. To find the total number of gifts your generous true love has given to you, set
the following:

That’s a pretty decent haul!

Return to the party

Back to the Birthday problem: in order to find out the number of possible combinations of birthdays shared between two people in a group of 23 people you need to know the 22nd
triangular number:

With this many pairings of 23 people, it now seems more reasonable that there is a 50% chance of two of them sharing a birthday.

To show the exact probability here, it is actually easier to find the opposite – the chance that
no
two people share a birthday – and exploit the fact that in probability the
chance of something happening and the chance of something not happening have a sum of 100%.

Alan’s birthday can fall on any of the 365 days of the year, leaving 364 alternative days on which Blaise’s birthday could fall. In turn, there are 363 days on
which Carl’s birthday could fall in order for it not to be shared with either Alan or Blaise. By the time the 23rd guest, Walter, steps into the room there are 343 possible days on which
his
birthday could fall without it being shared by anyone else in the room. If you write each one of these numbers as a probability out of 365, and then multiply them together, the total
probability generated is:

which means that the probability of two people sharing a birthday when there are 23 people in the room is 50.7%.

T
HE
A
LIENS
H
AVE
L
ANDED

In 1960 American astronomer Frank Drake (1928–) was the first person to use radio telescopes to search for signals, messages or other evidence of intelligent life in the
universe. This spawned what is now known as
SETI
– the Search for Extra-Terrestrial Intelligence.

Drake developed an equation to calculate the number of civilizations in the Milky Way that we should be able to communicate with:

R* is the number of new stars made in the galaxy each year; f
_p
is the fraction of new stars that will have planets; n
_e
is the number
of potentially life-supporting planets; f
_l
is the fraction of the life-supporting planets that are known to have life on them; f
_i
is the fraction of planets that have
intelligent life; and f
_c
is the fraction of planets that emit some kind of evidence of their civilization, such as radio waves. L represents how long such evidence emits for.

At the time of writing, many of these factors are pure conjecture, and scientists have come up with widely varying answers. Why not try your own!

The Future of Mathematics

We’re not done yet. Of course, the use of new computer methods to solve numerical problems that were previously deemed impossible to solve have boosted enormously
developments in technology, science, medicine and engineering. However, there still remain thousands of unsolved problems in mathematics and science that will keep the experts busy for some time to
come...

T
HE
M
ARCH
F
ORWARD

In 1900 at the International Congress of Mathematicians German mathematician David Hilbert (1862–1943) posed twenty-three mathematical problems that he felt were key to
the development of the subject. Since the congress ten of Hilbert’s problems have been solved, seven have been solved to some extent or have been shown not to have a solution, three were too
vague to be solved and three remain unsolved.

Posing these problems had exactly the effect that Hilbert wanted – the competition spurred mathematicians to strive to tackle them and in the process forge into new
areas of research. In 2000, in much the same vein as Hilbert, the Clay Mathematics Institute issued another seven problems, now known as the Millennium problems.

So far, only one of the problems has been solved – the Poincaré conjecture, which relates to the topology of spheres. It was solved by an extraordinary Russian mathematician called
Grigori Perelmann, who has declined not only a Fields Medal (the highest accolade in mathematics) but also the $1 million dollar prize from the Clay Institute.

One of the problems posed in both Hilbert’s problems and the Millennium problems is the Riemann hypothesis, a problem that many mathematicians feel is the most important in mathematics. It
concerns the distribution of prime numbers. The Goldbach conjecture (see
here
) tells us approximately where the prime numbers should be; the Riemann hypothesis would help us to know how far
away from the expected place the prime should actually be.

W
HAT
N
EXT
?

The future of mathematics depends very much on mathematicians who are, as I write, children, or as yet unborn. In order to cultivate the best possible mathematicians and
scientists to help solve the world’s problems we need people with excellent mathematical training, which is quite an educational investment. In our current educational system, every
schoolchild is taught
numbers and arithmetic through to algebra and geometry so that by the onset of adulthood they have the tools necessary to enter a technical career path,
should they so choose.

The majority of people, however, do not enter a technical career and therefore do not necessarily need mathematics taught beyond primary school. Most people use calculators or, more frequently,
mobile telephones with built-in calculators, to do the mundane arithmetic that is all the maths needed in everyday life.

So, should we continue to make mathematics a compulsory subject until the age of sixteen? There are clearly those who enjoy maths and those who do not. Perhaps we could just teach basic
arithmetic and everyday maths to younger children and save the harder, more interesting stuff as an optional course for older children who show a particular inclination and aptitude towards the
subject? Well, if it worked for the Ancient Greeks...