## Archive for November 11th, 2012

### A simple way to calculate the day of the week for any day of a given year

Sunday, November 11th, 2012

Image: Jeremy Church

The other day I read a tweet about how someone was impressed that a friend had been able to tell them the day of the week given an arbitrary date.

There are a bunch of general methods to do this listed on the Wikipedia page for Determination of the day of the week. Typically, there are several steps involved, and you need to memorize some small tables of numbers.

I used to practice that mental calculation (and many others) when I was about 16. Although all the steps are basic arithmetic, it’s not easy to do the calculation in your head in a couple of seconds. Given that most of these questions that you’re likely to face in day-to-day life will be about the current year, it seemed like it might be a poor tradeoff to learn the complicated method to calculate the day of the week for any date if there was a simpler way to do it for a specific year.

The method I came up with after that observation is really simple. It just requires you to memorize a single 12-digit sequence for the current year. The 12 digits correspond to the first Monday for each month.

For example, the sequence for 2012 is 265-274-263-153. Suppose you’ve memorized the sequence and you need to know the day of the week for March 29th. You can trivially calculate that it’s a Thursday. You take the 3rd digit of the sequence (because March is the 3rd month), which is 5. That tells you that the 5th of March was a Monday. Then you just go backward or forward as many weeks and days as you need. The 5th was a Monday, so the 12th, 19th, and 26th were too, which means the 29th was a Thursday.

It’s nice because the amount you need to memorize is small, and you can memorize less digits if you only want to cover a shorter period. The calculation is very simple and always the same in every case, and you never have to think about leap years. At the start of each year you just memorize a single sequence, which is quickly reinforced once you use it a few times.

Here’s Python code to print the sequence for any year.

``` #!/usr/bin/env python import datetime, sys try:     year = int(sys.argv[1]) except IndexError:     year = datetime.datetime.today().year firstDayToFirstMonday = ['1st', '7th', '6th', '5th', '4th', '3rd', '2nd'] months = ['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun',           'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec'] summary = '' for month in range(12):     firstOfMonth = datetime.datetime(year, month + 1, 1).weekday()     firstMonday = firstDayToFirstMonday[firstOfMonth]     print months[month], firstMonday     summary += firstMonday[0] print 'Summary:', '-'.join(summary[x:x+3] for x in range(0, 12, 3)) ```

The output for 2012 looks like

```Jan 2nd
Feb 6th
Mar 5th
Apr 2nd
May 7th
Jun 4th
Jul 2nd
Aug 6th
Sep 3rd
Oct 1st
Nov 5th
Dec 3rd
Summary: 265-274-263-153
```

The memory task is made simpler by the fact that there are only 14 different possible sequences. Or, if you consider just the last 10 digits of the sequences (i.e., starting from March), there are only 7 possible sequences. There are only 14 different sequences, so if you use this method in the long term, you’ll find the effort of remembering a sequence will pay off when it re-appears. E.g., 2013 and 2019 both have sequence 744-163-152-742. There are other nice things you can learn that can also make the memorization and switching between years easier (see the Corresponding months section on the above Wikipedia page).

Here are the sequences through 2032:

```2012 265-274-263-153
2013 744-163-152-742
2014 633-752-741-631
2015 522-641-637-527
2016 417-426-415-375
2017 266-315-374-264
2018 155-274-263-153
2019 744-163-152-742
2020 632-641-637-527
2021 411-537-526-416
2022 377-426-415-375
2023 266-315-374-264
2024 154-163-152-742
2025 633-752-741-631
2026 522-641-637-527
2027 411-537-526-416
2028 376-315-374-264
2029 155-274-263-153
2030 744-163-152-742
2031 633-752-741-631
2032 521-537-526-416
```