The mathematical muddle created by leap years

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I have a friend at work – in the mathematics department of the University of Bath in the UK – who is turning 11 this year. He’s not a child prodigy (although we definitely do get some of those in maths). He just has a very special birthday: February the 29th.

As 2024 is a leap year, it means he gets to celebrate on the actual date of his birth instead of one of the surrounding days. Although for my colleague it is undoubtedly tedious to have people like me joke about how old he is (and spare a thought for the 100 year old “leaplings” who have had to endure 25 such occasions), for the rest of us the leap year has a special, almost mystical, aura about it.

This exceptional day has been associated with all sorts of weird and wonderful traditions over the years: from the wildly outdated notion that 29 February is the only day when women can propose to men, to the Leap Year Festival held in Anthony, New Mexico, which sees people born on this special day gather to celebrate their rare birthdays together.

As a rule of thumb, leap days come around every four years. But there are exceptions to this rule. For example, at the turn of every century we miss a leap year. Even though the year is divisible by four, we don’t add a leap day in the years that end in 00. But there’s an exception to this rule too. If the year is a multiple of 400 then we do add in an extra leap day again. At the turn of the millennium, despite being divisible by 100, the year 2000 did, in fact, have a 29 February because it was also divisible by 400.

So far so complicated. But why do we have leap years at all? And why are the rules that govern them so convoluted? As you probably know, the answer is something to do with keeping things in sync.

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