Calendar Math

What day of the week will May 12, 2034 be?

What day of the week was May 12, 1298?

The following algorithm will tell you.

*(Note: all divisions, except where noted otherwise, are integer divisions, in which remainders are discarded.)*

First figure out the values for \(a\), \(y\), and \(m\) -- variables to be plugged into a formula.

\(a = \frac{14 - month}{12}\)* (month = # of month, 1 for Jan, 2 for Feb, etc)*

\(y = year - a\) *(year = the 4 digit year)*

\(m = month + 12a - 2\)

Next, plug the values of y and m into the following formula to calculate the day:

\[ d = (day + y + \frac{y}{4}-\frac{y}{100}+\frac{y}{400}+ \frac{31m}{12}) \:mod \:7 \]

*(Note: mod 7 means "modulo division." That is, take the remainder instead of the quotient as your answer. For example, 20 mod 3 = 2, because the remainder is 2.)*

The answer you get for \(d\) will correspond to a day of the week as such:

0 = Sunday1 = Monday

2 = Tuesday

3 = Wednesday

4 = Thursday

5 = Friday

6 = Saturday

**Example:** What day of the week will April 5, 2020 fall on?

First figure out \(a\), \(y\), and \(m\):

\(a = \frac{14-4}{12} = 0\)

\(y = 2020 - 0 = 2020\)

\(m = 4 + 12(0) - 2 = 2\)

Now plug \(y\) and \(m\) into the \(d\) formula to calculate the day:

\[ d = (5 + 2020 + \frac{2020}{4} - \frac{2020}{100} + \frac{2020}{400} + \frac{31(2)}{12}) \: mod \: 7\] \[ d = (5 + 2020 + 505 - 20 + 5 + 5) \: mod \: 7 \] \[ d = 2520 \: mod \: 7 \] \[ d = 0\]*Note: 2520/7 = 360 with a remainder of 0.*

Recall from above that 0 = Sunday. So April 5, 2020 will be a Sunday.

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