Everyone knows that there are 365 days in a year — except on a leap year, when there are 366. The reason an extra day is added to the calendar every four years is because the number of days in the year is actually 365.2422. Now, 0.2422 roughly equals 0.25, so the year is approximately 365 and one quarter days long. If we did not have leap year, then the relative postition of the earth and the sun would change slightly each year. Eventually, Boston's summer would start in January.

1. Adding one day every four years to the calendar makes the average calendar year 365+(1/4) days long. This is just little longer than the actual length of 365.2422 days. To compensate, leap year is skipped in years ending with 00 (that is, every hundred years). How close is the new average to 365.2422? Express the average length year as both a decimal number and sum and difference of fractions.
2. The correction given above is pretty good, but still not perfect. One way to improve it is to, every so often, have a year ending in 00 (which would usually not be a leap year by the above) be a leap year. How often should this be done? How close is the new average calendar year to the actual year? Express the average length calendar year as both a decimal number and a sum and difference of fractions. (Hint: The year 2000 was a leap year.)
3. Can this process be continued to make the average length calendar year even closer to the actual length? What happens?
4. The process used above has the advantage that it is easy to remember. Could you come up with a more accurate calendar using the same idea (skipping or not skipping leap years in a regular pattern)?

The math question behind setting up a system of leap years is this--Can a decimal number (like 0.2422) be approxated by a sum and difference of fractions with numerator 1 and denominators forming a regular sequence. Is this always possible? Try this with other decimals (e.g., 0.618034, or your favorite decimal between 0 and 1). Are there other ways of representing a decimal as a sum of fractions that could be used to make a reasonable calendar?

Communicated by the Chelsea High School Mathematics Department.