Year 2^n has 29 - n days, and year 3^n has 29 + n days. Hence, February can be of any integer length on Planet Keith. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

----- Original Message ----- From: Keith F. Lynch Sent: 02/09/13 10:33 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Weird leap-year rule

In our calendar, February has 28 days unless the year is divisible by 4, in which case it has 29, unless the year is divisible by 100, in which case it has 28, unless the year is divisible by 400, in which case it has 29.

But there's a world about a trillion light years away that by a remarkable coincidence has a calendar almost identical to ours. The one difference is their leap year rules. As with us, their February starts with 28 days. But it gains a day for every odd number that the year is divisible by, and loses a day for every even number that the year is divisible by. Divisors include 1 and the number itself.

So in the year 1, February has 29 days, since 1 is divisible by just one odd number, namely itself, and is not divisible by any even numbers. So year 1 has 366 days. February of the year 2 has 28 days, since 2 is divisible by one odd number (1) and one even number (2). So year 1 has 365 days.

Like us, they never had a year zero, but unlike us they have a good excuse for the lack: Since zero is divisible by infinitely many odd numbers and infinitely many even numbers, it would be impossible to decide on February's length.

Eventually, they have a crisis. For the first time, February will have a negative length, and there's no provision for dealing with that. Borrow a day from January or March? The puzzle is, in what year does this first happen? Or will this actually never happen?

Suppose they do decide to borrow days from other months when February has a negative length. What then will be the first year in which a whole year has a negative length? Or will this actually never happen?

Is there any (integer number of days) length, positive, negative, or zero, that a year can never have?

Finally, and most interestingly, what is the average length of a year in their calendar? If possible, express it in closed form rather than as a numerical estimate.

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