## Monday, September 04, 2006

### Sum Days

Some people believe that January 1, 2000 is the first day of the 21st century. Other people believe that the honor belongs to January 1, 2001. But everyone should agree that January 1, 2002 is the first "sum-day" of the new century- when you write out that date in standard notation, it becomes 01/01/02, and 1+1=2. More generally, a sum-day is a date in which the day and month add up to the year. With that in mind:

A) What is the last sum-day of the 21st century?

B) How many sum-days are there in the 21st century?

1. Well, we have January 1, 2002 and January 2 2003 and .... pick a date, I'll name the year. September 4 2013. May 1 2006. July 14 2021. Every day of the year has a "sum day" sometime in the century.

And February 29? Noooo. 2/29/31 doesn't exist, it's not a leap year. So the total is 365 sum days.

And the last one? Highest day, highest month: December 31 2043.

2. Looks like this was too easy for you, Jonathan. Nice job.

A) The last sum-day of the 21st century is December 31, 2043, because 12+31=43, and both the month and day are as big as possible.

B) This one is much easier than you might think. The correct answer is 365, because every day in a standard (non-leap) year is part of a sum-day for some year. For example, November 26 is a sum-day for the year 2037, because 11+26=37. The only date for which this principle doesn't work is the leap day, February 29. That's because 2+29=31, but 2031 is not a leap year.

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