Problem:

Example: What day did July 9, 1957 fall on?

Answer: Tuesday

Solution:

We know that in the year 1900, April 4 was a Wednesday.

This means that the day 4/4/1900 was a Wednesday. (THIS IS THE KEY TO THE PROBLEM)

Also, in 1900, the following days were all Wednesdays: June 6, August 8, October 10, and December 8. Each of these days are as important as 4/4. When asked what day you are looking for, you need to know which day of the month was a wednesday. Well, if you notice, these days are 4/4, 6/6, 8/8, 10/10, and 12/12 when written in MM/DD form. There are four other days that we need to concern ourselves with that are equally important to 4/4: 5/9, 9/5, 7/11, and 11/7. Figure out your own method to remember a pattern. All of these days are Wednesdays. Also, notice that we have only dealt with the months of April through December. January through March requires a bit more guile.

Given this information, we are given a date to find the day of the week, say the example from above, July 9th, 1957.

• First, take the last two digits of the year we are looking at: 57.
• Take this number and divide by 12 (this 12 represents the number of months in a year).
• The answer is 57/12 is 4 with remainder 9.
• Next, find out how many leap years can occur in remaining 9 years (from the remainder above). Obviously 9/4 is 2 leap years.
• Add 4 (the quotient) + 9 (the remainder) + 2 (the number of leaps years in the remainder).
• Your sum is 15; now take this sum and make it congruent to __ (mod 7). We know that 15 is 1 (mod 7).
• Take this 1 from 1(mod 7) and add 1 day of the week to 7/11/1900 (Wednesday).
• Now, it becomes Thursday
• We are looking for July 9th, though, so if July 11, 1957 is a Thursday then 2 days prior to that is a Tuesday.

Let's take a look at a second example. Suppose we want to find out what day of the week December 7, 1941 ('a day that will live in infamy'; maybe, you've heard of it). We will repeat the same process from above.

• Find the last two digits of the date: 41
• Take this number and divide by 12: 41/12 = 3 with remainder 5
• The number of leap years that can occur in the remainder: 1 leap year
• Add 3 + 5 + 1 = 9
• Take this sum and make it congruent to __ (mod 7); 9 is congruent to 2 (mod 7);
• Use the 2 from the 2 (mod 7) and add 2 days to Wednesday.
• The day that December 12, 1941 fell on was a Friday (this is why 12/12 is one of our key days).
• This means that December 7, 1941 was a Sunday.

Here's a date where our work gets a bit difficult: Find the day of the week that Valentine's Day fell on in 1903; so we need to find the day of 2/14/1903.

• The last two digits: 03
• 03/12 = 0 remainder 3
• 0 leap years occur in the remainder.
• Adding 0 + 3 + 0 = 3
• 3 is congruent to 3 (mod 7).
• Here's the problem: We have no day in February determine the Wednesday. We do know that we can add 3 to either April 4 and count backward or add 3 to December 12 add count forward...nothing ingenious; just counting!
• From the last statement, April 4, 1903 was a Saturday as was December 12, 1902 a Friday.
• Counting forward from December 12, 1902: December 19 and 26th were Fridays; so January 1, 8, 15, 22, 29 were Thursdays; January 30 was a Friday, and the 31st was a Saturday; so February 1 was a Sunday; the 8th was a Sunday, as was the 15th; so Valentine's Day (the day before) was a Saturday.
• Counting backward (which I'm not as good): April 4, 1903 was a Saturday; so was March 28th. From this the 21st, 14th, and 7th, were all Saturdays. February 28th (since only years that are multiples of 4 are leap years) was a Saturday, as were the 21st and 14th, Valentine's Day!

The question now becomes, "Can we determine days from different centuries?" This may be a solution for the reader to determine, but from what we've determined, April 4 of the year 2000 is a Thursday. So all of the key dates that we have to keep track of a Thursdays for the 21st century. Using a bit of the logic we have developed, we can assume that the key dates for the 19th century are Tuesdays, and the key days for the 18th century are Mondays.

So if anyone asks you what day of the week July 4, 1776 fell on, remember Monday is the key day; and 4/4, 6/6, 8/8, 10/10, 12/12, 5/9, 9/5, 7/11, and 11/7 are the key dates.

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