I’m not greatly interested in cars, but I am beginning to think about buying a new one. Our present car has a registration number that begins AU04. Anyone reading this in the UK should be able to work out from that roughly how old it is. The 04 here indicates that the card was first registered in the first half of the year 2004. So it’s getting on a bit – like its driver.

The system for the two digits in the registration is such a clever but simple idea. Once it was decided that (a) 2 digit numbers would be used, and (b) that the code would change every six months, it follows that the system can run for 50 years in total – because there are only 100 two-digit codes available: 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12 … and so on, up to 99.

To use these in that order would have been a standard example of ordinal numbers: using numbers to label things and to put them in order. So, for example if we were up to 23 we would know that the next code to be used would be 24.

But the clever idea was to use a different ordering system with a third requirement: (c) the first six months of each year should correspond to the year itself.

So, the first 50 codes (00 to 49) cover the first six months from the year 2000 to the year 2049. Then the remaining codes cover the second six months in each of these years, giving this sequence:

00, 50, 01, 51, 02, 52, 03, 53, 04, 54, 05, 55, 06, 56, 07, 57, 08, 58, 09, 59, 10, 60, 11, 61, 12, 62, 13, 63, and so on.

So, if I buy a new car now (in May 2011) it will have the registration code 11 (indicating the first half of 2011). But if I buy it in August then it will have 50 added to this to give a registration code of 61.

To work out the year of registration, you use either the two-digit code or the two-digit code minus 50 if the number is greater than 49. For example, 35 will be the first half of the year 2035; and 85 will be the second half of the year 2035 (85 – 50 = 35).

The sequence of codes, starting with 00, is generated by the following pattern of rules: add 50, subtract 49, add 50, subtract 49, add 50, subtract 49 … repeated over and over again, until all the 2-digit codes are used up.

A simple, but pleasing bit of mathematics.

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