Let's answer the second question first.

Why is March 14 p Day?

Because if we write the date as 3.14 (this is how some people write 3/14) we have the first two places of p. Seems like 3.15 (the 15th of March) would be closer... In the future we have March 14, 2015 to look forward to: it might be written 3.14.15 which gives the first 4 places of p.


Now on to the first question.

Let me Count the Ways. . .

Lisa Hoffman said "Marriage is like p--natural, irrational and very important." Some might draw the analogy between love and p, too, in that both go on forever. Valentine's Day just passed, and every year this holiday is celebrated on February 14. Math folks celebrate p Day, which occurs on March 14. (Admittedly, fewer cards are sent on p Day than on Valentines' Day.)

On average, how many hours pass between Valentines' Day and p Day?

The solution posted in the MathCounts column is 654. This actually is WRONG! However, it's close enough for most people, so we'll call it "correct."

Starting at the end of February 14 -- right at midnight -- and going to the beginning of March 14 -- right at midnight -- is a period of

Most people think leap year occurs 1 in every 4 years (more on this below). So, in every 4 years we have 648 hours between 3 times and 672 hours between 1 time. The average is then (648 + 648 + 648 + 672)/4 = (3*648 + 672)/4 = 654 hours.

You might think of it in terms of the following "probability" table (although, occurence of a leap year is strictly deterministic).

x 648 672
p(x)    0.75    0.25

The mean of this "distribution" is (almost all statistics books have a formula for this)

m = 648*0.75 + 672*0.25 = 654

(it's no more than a weighted average of 648 and 672).

(Historical aside. This page was created in the mid-1990s. In 2007, I learned that my initial correction to this answer was incorrect. Ayo, a med student at Cambridge University, supplied the corrective, as well as some interesting source material (see below). I've self-inflicted the appropriate penalty: I will forgo my pudding this evening. To be frank: I appreciate very much that people are looking at this stuff. I should look at it more often - I found and corrected or deleted a number of holes, errors, and typos.)

This answer (654) is incorrect because leap year does occur in years divisible by 400: 400, 800, 1200, 1600, 2000 (remember?), etc. Adapt the analysis to sets of 400 consecutive years. Out of every 400 years there are 100 that are divisible by four, but 3 of these are leapless, for a total of 397 leap years. Then there are 303 leapless years. The average is then

(303*648 + 97*672)/400 = 653.82 hours.

That's 27 days, 5 hours, 49 minutes, 12 seconds.


Sources

Math Forum

Earth/Matrix: Science in Ancient Artwork

Wikipedia