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Malhar Rajpal In my previous mental maths article I outlined a technique to quickly square two-digit numbers in your head, which is an invaluable skill in maths competitions. Today, we will look at a trick to quickly solve a familiar type of question: finding the day of the week for any given date in the 21st century. This article will describe the method while my next article will explain why it works. The method itself is quite straightforward. It follows the formula: Day of week code = Remainder of [(Date + Month Code + (Last two digits of year + 6) + Quotient of [Last two digits of year / 4]) / 7] This may look confusing at first but let’s break it down: Date: The number for date is simple, it’s simply the day of the month. For example during April 27th, the date would be 27 and during December 9th, the date would be 9. Month Code: The month code is a single digit number that is associated with every month of the year. I have listed the codes below in a table and the reasoning behind each of these codes will be explained in the next article. This will require a little bit of memorisation and I recommend using some kind of key word or phrase to help you remember each month's code, it’s not that difficult with some effort! *On a leap year, the month codes for January and February become 6 and 2 respectively(1 less than their usual code) Remember that leap years occur on every year that is divisible by 4 except end of century years that are not divisible by 400. So 1904 and 2000 are leap years, yet 1900 is not because although it is divisible by 4, is an end of century year and isn’t divisible by 400. (Last two digits of year + 6) + Quotient of [Last two digits of year / 4]: The last two digits of year section is pretty self explanatory: for example if the year is 2024, the last two digits are 24, or if the year is 2001, the last two digits are 01. Make sure you add the +6 to the last two digits of the year - that is necessary to make the calculation correct. The quotient of [last two digits of year / 4] is simply the answer to the division problem of the last two digits / 4 without the remainder. So if the year is 2027, The quotient of [last two digits of year / 4] is the integer part of 27/4 = [6.75] = 6, and if the year is 2085, the quotient of [85 / 4] is 21. Putting it all together: Let’s take a random date in the 21st century, 6th May 2045, and try to find the day of the week. Using the formula: Day of week = Remainder of [(Date + Month Code + (Last two digits of year + 6) + Quotient of [Last two digits of year / 4]) / 7], We quickly see the date of May 6th 2045 is simply 6, the month code for May (referring to the table above) is 1, the last two digits of the year + 6 is simply 45 + 6 = 51, and the quotient of [last two digits of year/4] is [45/4] = 11. Plugging all of these numbers into the formula: Day of week code = Remainder of [(6 + 1 + 51 + 11)/7] = Remainder of [(69/7)]. The remainder of 69/7 is 6 since 69/7 is 9 with 6 leftover. And thus we get day of week = 6. What does this mean? To find out the day of the week, we follow this table: Since we got 6 as a remainder, we can use our table to figure out that the associated day of the week is Saturday. Hence, we can confidently say that May 6th, 2045 will occur on a Saturday!
Try it for yourself, choose a random date in the 21st century and try out the method! In the next article, I will prove why this formula works.
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