Quickly Finding the Day of the Week

Date: 11/14/2000 at 12:37:16
From: V Anand
Subject: Day vs date

Dear Dr. Math,

I have a simple question.

Today is November 14, 2000, a Tuesday. What day of the week was 
November 14, 1901?

I got this question in a competitive exam where they expect an answer 
in 30 seconds. I have seen your calendar chart in the archives, but is 
there a quick method?

Thanks,
Anand

Date: 11/14/2000 at 13:03:15
From: Doctor Rick
Subject: Re: Day vs date

Hi, Anand.

Here's what I would do if I only had 30 seconds (and I don't know that 
I'd make it):

That's a difference of 99 years. How many leap years in that time? 
99/4 = 24. Add the 24 leap years to the 99 years to get 123. Divide 
123 by 7 and get 17, remainder 4. Go back 4 days from Tuesday, and I 
get Friday.

The day of the week for a particular date advances by 1 day every 
non-leap year, and 2 days every leap year. That's why I added the 
number of leap years to the number of years elapsed. Then, when the 
day advances by 7 days, it's back where it started; so I took the 
remainder on division by 7. Finally, I worked backward to what day it 
would have to be in order to advance by 4 days and get to Tuesday.

I wouldn't have time to stop and think about whether I had properly 
counted the leap day in 2000, so I wouldn't be quite sure (without 
further thought) that I had it right. As it turns out, I haven't: I 
could have seen this by partially listing the leap years between 1901 
and 2000: 1904, 1908, ..., 2000. The numbers 4, 8, ..., 100 are 4 
times the numbers 1 to 25, so there were 25 leap years. I have to go 
back one more day, to Thursday. I think this is correct. I've missed 
the 30 seconds, but we've both learned something.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/