You might be surprised to hear that the questions you have been playing with here take you right to the limits of modern maths. For centuries mathematicians have been asking themselves a question that is very similar to our problems involving coins and cinema tickets: Given a whole number n, say n = 3, in how many ways can you write n as a sum of other whole numbers?
For n = 3 the answer is easy:
3 = 1+1+1
3 = 2+1
3 = 3+0.
So there are three ways of writing 3 as a sum of other whole numbers.
What about n = 100?
You could try writing down all the different possibilities, but we wouldn't advise it: it turns out that there are nearly 200 million ways of writing 100 as a sum of whole numbers!
It would be useful to have a neat formula which, for any whole number n, gives you the number of ways in which you can write n as a sum. That number is called the partition number of n. Such a formula would save you the effort of writing down all the different possibilities and counting them.
For over 300 years mathematicians have been looking for such a formula. They have found many different ones, often using very complex maths, but there is a problem: using these formulae to work out the partition number of very large n still involves so many calculations, it's impossible to do even on a very fast computer. So even now we don't know the partition numbers of very large n.
The largest partition number that has been computed so far (on March 2, 2014) counts the number of ways in which you can write
as a sum of whole numbers.
The answer is a number that has more than ten billion digits. It starts with
and ends with
See here to find out more.