M^3 Soln (Monday Math Madness Solution)

Monday Math Madness is over again. I got the right answer to the problem, but I didn’t win the prize (sigh). The solutions were all about the same, so I’m posting what I emailed below the fold…

Here’s my solution to this week’s question [http://wildaboutmath.com/2008/03/31/its-monday-math-madness-time-contest-3/]:

Date: Day of the year in a normal year (day of the year in a leap year)

Jan 16th: 16th day of a NY (16th day in a LY)
Feb 16th: 47th day of a NY (47th day in a LY)
Mar 16th: 75th day of a NY (76th day in a LY)
Apr 16th: 106 day of a NY (107th day in a LY)
May 16th: 136 day of a NY (137th day in a LY)
Jun 16th: 167 day of a NY (168th day in a LY)
Jul 16th: 197 day of a NY (198th day in a LY)
Aug 16th: 228 day of a NY (229th day in a LY)
Sep 16th: 259 day of a NY (260th day in a LY)
Oct 16th: 289 day of a NY (290th day in a LY)
Nov 16th: 320 day of a NY (321th day in a LY)
Dec 16th: 350 day of a NY (351th day in a LY)

Now we do a little work with modular arithmetic (the remainder when divided by some number). There are 7 days in a week. So let’s see what we come up with if we take these numbers mod 7 (the remainder when divided by 7):

Normal YearLeap Year
16 mod 7=216 mod 7=2
47 mod 7=347 mod 7=3
75 mod 7=576 mod 6=6
106 mod 7=1107 mod 7=2
136 mod 7=3137 mod 7= 4
167 mod 7=6168 mod 7=0
197 mod 7=1198 mod 7=2
228 mod 7=4229 mod 7=5
259 mod 7=0260 mod 7=1
289 mod 7=2290 mod 7=3
320 mod 7=5321 mod 7=6
350 mod 7=0351 mod 7=1

Let’s interpret this. Since each of those numbers (0,1,2,3,4,5,6) represent days of the week, then in a normal year, we see all the digits (0-6) appear. Hence, not only will there be a Friday the 16th, but also a Monday the 16th, a Tuesday the 16th, a Wednesday the 16th, a Thursday the 16th, etc. It’s an even stronger result! Cool! The same thing can be said for a leap year too.

Note: Just for a bit of explanation of the modular arithmetic and how it relates to the day of the week, let’s go really slowly and think about the month of January.

January 1st is the 1st day of the year.
January 2nd is the 2nd day of the year.
January 3rd is the 3rd day of the year.

January 31st is the 31st day of the year.

Let’s do our modular arithmetic.

1st day of the year: 1 mod 7=1
2nd day of the year: 2 mod 7=2
3rd day of the year: 3 mod 7=3

7th day of the year: 7 mod 7=0
8th day of the year: 8 mod 7=1
9th day of the year: 9 mod 7=2

31st day of the year: 31 mod 7=3

So if January 1st is a Tuesday, for example, then 1 represents Tuesday, 2 represents Wednesday, 3 represents Thursday, 4 represents Friday, 5 represents Saturday, 6 represents Sunday, and 0 represents Monday.

So, to continue our example, we know that January 8th is a tuesday (obviously) and January 31st is a Thursday (since it has a remainder of 3). And the 249th day of the year will be Friday (because 249 mod 7=4).

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