You might have heard of the seven Clay Mathematics Institute Millenium Prize Problems, each of which offers a US$1,000,000 prize for its solution. Given that mathematics is sometimes done with nothing more than paper, pencil, persistence and perhaps a bit of lateral thinking, any one of these problems might seem to be an easy route to riches. But of course the problems are ones that have already resisted the onslaught of many great minds. Depending on your own mathematical experience, you will probably find that these problems fall into three groups:

• those that you understand what the problem is,
• those where you understand how much you would have to study to understand what the problem is, and,
• the "mass gap" problem.

If you want to spend your time on one of these problems, then good luck to you, and don't let me stop you. But if you want to know about a mathematical and scientific puzzle that has not yet had the big guns of mathematics and science properly brought to bear on it, and which could easily make you a billion dollars if you solve it, then read on ...

The Puzzle

As you might have guessed, if you came to this page from elsewhere on this site, the puzzle to be solved is the puzzle of what music is.

Before I go on to explain why you should think this is a puzzle that you might easily solve yourself, I must justify the claim that you could get a billion dollars for solving it. I need to make it quite clear that I don't have a billion dollars to give to you if you solve the problem (unlike the Clay Mathematics Institute, who presumably have $7,000,000 sitting somewhere, waiting for future winners to collect it).

If you solve the problem, it will be up to you to use the solution to make yourself the billion dollars. As I point out in my book, What is Music?: Solving a Scientific Mystery, when I discuss the "Luxury Yacht Test" for a theory of music, this should not be so difficult. If you solve the mystery of what music is, you should be able to use that information to create a generative algorithm for musical composition, and you should be able to use that algorithm to generate commercial quality music, which you could then sell in the normal way to receive royalties. I cannot absolutely guarantee that you would receive as much as $1,000,000,000, but to get that much you would only have to receive – in return for composing music better than any music anyone has ever heard before – $1 from each of the 1,000,000,000 richest people in the world (which is about the size of the world's "middle class").

Why the Puzzle Might be Easy to Solve

It's one thing to say that this or that problem represents a billion dollar business opportunity – it's another to suggest that anyone could plausibly solve such a problem.

The following is a list of reasons why I think that any intelligent person prepared to make a moderate effort has as good a chance as anyone else of making the final breakthrough:

• It's not hard to understand what the problem is. In fact the hardest thing is to realise that there is a problem, something that our everyday familiarity with music can hide from us. Chapter 2 of my book gives an exhaustive account of what "the problem" is and how we know that it really hasn't been solved yet.
• Although more and more professional scientists are taking an interest in the problem, I think it is fair to say that no one has a clue yet what the answer is. The best evidence for this is that there are so many different theories that are seriously proposed, including but not limited to:
  • Music is "auditory cheesecake" (Steven Pinker's theory): there happen to be various reasons why certain features of sounds are enjoyed by human listeners, and if you do enough of them at once then you have music.
  • Music has evolved for the purpose of attraction between the sexes.
  • Music has evolved for the purpose of general social bonding.
  • Music has evolved to help people learn stuff by rote.
  • Music is an extension of mother-child interaction.
  • Music has something to do with anticipation and expectation.
• A substantial portion of the available data about music is the music itself. This data takes a simple mathematical form, and it is easy enough to "experiment" on it with a computer or with any musical instrument. If necessary it can be analysed using just pencil and paper (sometimes a ruler helps as well).
• There appear to be opportunities to apply even quite basic concepts of abstract mathematics to music which the professional music science community has not yet taken. This is made particularly evident in Chapter 9 of my book, where I analyse all of the known symmetries of music, as I have not seen a similar analysis anywhere else. If you read it then you will discover, among other things, that the phenomenon of pitch translation invariance (actually a mathematical name for "music sounds much the same when played in a different key") raises fundamental questions about the nature and purpose of the information processing involved in music perception.
• A final reason comes from detailed consideration of the theory of constant activity patterns given in my book. If that theory is correct, then a full solution to the music problem consists of identifying all of the cortical maps relevant to the perception of music and understanding how constant activity patterns in those cortical maps give rise to corresponding features of music. I've already done some of the work – describing some of those maps – and all you have to do is find the rest. Of course if it was that easy then I would have done it myself, but often one person can easily see the solution to a problem that another person gets stuck on. And that person could be you.