## Thursday, 16 September 2010

### Anniversaries solution

Last week I posed this question:
Is it unusual that a 45th anniversary should fall on the same day of the week as the original event?

Here are my thoughts on this.

First, we note that in a year of 365 days an anniversary moves on by one day of the week. This is because 365 is a multiple of 7 (364) plus 1. So a Saturday moves on to a Sunday, and then to a Monday and so on.

However, when the year following the anniversary includes a 29th February then the anniversary moves on by two days of the week (for example, an anniversary on a Monday will be on a Wednesday the following year.)

Since leap years (normally) happen every four years, then in a period of four years there will be one 29th of February. So anniversaries move on by 5 days in every 4 years. This is the general rule you can now apply to most questions like this.

For example, because 45 = 1 + (11 x 4), then in 45 years, the anniversary would move on by 1 + (11 x 5) days = 56 days. Since 56 is an exact multiple of 7, then the 45th anniversary will fall on the same day of the week as the original event.

But, wait! If the first year after the original event contains a 29th February, then the anniversary will move on by 2 days in the first year, followed by 11 lots of 5 days, making 57 days in total. This means that the 45th anniversary will fall one day later in the week than the original event.

So, here's a summary:
(A) If the first year after the anniversary does not contain a 29th February, then the 45th anniversary will normally fall on the same day of the week as the original event.
(B) If the first year after the anniversary does contain a 29th February, then the 45th anniversary will normally fall one day later in the week than the original event.

Clearly, because leap years occur normally every 4 years, then (A) will happen in about three-quarters of the cases.

Why do I keep saying 'normally'? Well, the above will have to be modified if the period in question contained the year 1900, which was not a leap year, or will contain the year 2100, which also will not be a leap year. Because a year in terms of the earth's orbit of the sun is an awkward length of 365 days, 5 hours, 48 minutes and 47 seconds, adjustments have to be made to the regular pattern of leap years. So, a leap year is every 4 years, but not every 100 years, except when the year is a multiple of 400 (as in 2000). That pattern just about balances things up.

One final thought: if the original event was on the 29th February, then the concept of an anniversary becomes rather weird. My advice would be not to do anything significant (like being born) on 29th February, unless you enjoy being a special case.