As Moira Stewart keeps on reminding us from billboards and TV screens, the deadline for submitting your tax return by paper is looming large. But if you were thinking of cooking up some random expenses to include in your return then be warned that the Inland Revenue is armed with some clever mathematical facts to spot the people who are faking their figures.

Most people, for instance, would expect that if you take all the figures that have been submitted as legitimate expenses to the Inland Revenue then the first digit is equally likely to be any of the numbers from 1 to 9. But this turns out not to be true. The number 1 will occur as the first digit three times more often than you would expect.

This strange phenomenon occurs with a whole range of sets of data. If you take the first digit in the lengths of the world's rivers or populations of cities across Europe or the set of numbers entered into Google each week, the number 1 wins every time.

This bias towards number 1 was first observed in 1881 by an American astronomer called Simon Newcomb. At the end of the 19th century, scientists didn't have computers or calculators to do their computations but instead used a book of logarithm tables to speed up numerical calculations.

Newcomb was struck by how much dirtier the pages at the front of the book were than those at the end. A book of logarithms is arranged rather like a dictionary. The front of the book contains numbers beginning with the digit 1, the numbers at the end of the book start with the digit 9. So Newcomb believed that the dirt was a sign that the numbers scientists were looking up in the tables began with 1 more often than any other number.

The theory seemed ridiculous. How can you use dirty pages to suggest something so counter-intuitive? It wasn't until 1938 that another scientist, Frank Benford, who worked for the General Electric Company in the United States, rediscovered this strange phenomenon. But this time he backed up the theory with an analysis of more than 20,000 numbers ranging over scientific, geographic and astronomical data.

Sure enough, the number 1 was appearing as the first digit 30 per cent of the time, the number 2, 18 per cent of the time, trailing off to less than 5 per cent of the numbers beginning with a 9.

You have to be careful what data you apply the rule to. It tends to apply especially to data that allows you to change the units you are using. For example, what if you submitted your expenses in dollars? If you'd spread the numbers evenly in sterling then look at the effect of multiplying all the expenses by 2. All the numbers beginning with 5, 6, 7, 8 or 9 have gone to numbers beginning with 1. This provides a hint as to why 1s might be more common in certain types of data.

Another indication of why the number 1 occurs more often comes if you look at house numbers as you walk down a long road. The first 20 numbers have a huge bias towards numbers beginning with a 1. As you get farther down the road this smooths out.

Until you hit the hundreds, that is, and then again 1 will begin to dominate. But a rigorous explanation for this phenomenon, now called Benford's Law, was provided only ten years ago by the American mathematician Theodore Hill.

If you're still not convinced, and have nothing better to do, why not write down all the numbers appearing in The Times today and check that they satisfy Benford's Law. And if they don't, then maybe a letter to the Editor is in order: is the newspaper faking its figures? And remember, when filling out your tax returns over the next few weeks, the Inland Revenue too is looking out for number 1.

Marcus du Sautoy's series The Story of Maths concludes on Monday on BBC Four at 9pm.

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