Science: Serendipity How not to fake a tax return

WITH the tax year coming to an end, I thought I would recount the tale of a serendipitous discovery which is being used by tax officials to help spot fraudulent tax returns. The story starts in 1938, when the American physicist Frank Benford was consulting a book of logarithm tables (before electronic calculators, log tables were used to perform calculations).

For example, to do a calculation involving the area of the United Kingdom (244,493km2), you would begin by looking up this number in your log tables. In fact, the tables only cover numbers from 1 to 10, and so you have to look up 2.44493 and then compensate appropriately.

When Benford picked up his book of tables, he noticed that the early pages were much more worn than the later ones. This was extraordinary, because it implied that the numbers that he was looking up were more likely to begin with 1 than with any other number, as opposed to the natural assumption that as many numbers start with 1 as start with 2, 3, 4, etc.

To see if numbers beginning with 1 were indeed more prevalent than other numbers, Benford checked a sample of 20,229 numbers from a wide variety of sources (lengths of rivers, baseball statistics, and so on). Sure enough, 30 per cent of the numbers began with 1, about 18 per cent began with 2, and the pattern continued - less than 5 per cent of the numbers began with 9. So if you open this magazine at random and find a number in an article, then there is a 30 per cent chance that its first digit will be 1.

Explaining the prevalence of 1 as the first digit (now known as Benford's Law) is not simple, but the following example illustrates part of the reason. Imagine, you have pounds 100 in a bank account. It earns 10 per cent annual interest, and you receive a statement each year. The first statement will read pounds 100, the second pounds 110, and the next statements would read pounds 112, pounds 133, pounds 146, pounds 161, pounds 177, pounds 195, pounds 214, pounds 236, pounds 259, pounds 285... up to pounds 814, pounds 895, pounds 985, pounds 1,083 and so on.

The first eight numbers begin with 1, but only four numbers begin with 2; only two begin with 8, only one begins with 9; and then we are back to numbers that begin with 1. In other words, numbers lazily wander through the 1s, but dash through the 9s.

Benford's Law is now being exploited by Californian tax officers. If somebody were to attempt to fabricate the numbers on their tax return, they would be tempted to ensure an even spread of numbers beginning with 1, 2, 3 and so on (which we know should not happen). So tax officials are checking the numbers in tax returns to see whether or not they deviate from Benford's Law.

If the numbers seem suspicious, this is not proof of fraud, but it does indicate that the return should be examined in detail.

Simon Singh is the author of `Fermat's Last Theorem' (Fourth Estate)