IS IT POSSIBLE TO DETERMINE WHAT DAY OFTHE WEEK ANY DAY WILL FALL ON?

The calendar has always been used as a simple tool, and today it comes pre-installed on all of our electronic devices. We use it to plan events in the future, set deadlines or know what weekday a certain date falls on. The calendar is simple. It is almost too simple, to the point where one may think that there is no reasoning, mathematics or algorithms behind how it works. These may be difficult to notice at first, but they’ll be described below.

After reading the title of this article, many people might think the answer is obvious: of course it’s possible to determine the weekday of any date — you just open the Calendar app. They might even assume that there is no other way. However, in this article I’ll explain a quick mental method that allows you to determine the day of the week for any date in the future - no phone, no internet and no guessing required.

This method is actually much less complex than one may think and involves only a few simple steps. These are the following (example: let’s imagine that someone asked what day of the week September 13th 2043 will be):

1) Choose memorable reference dates:

You must select a set of six to ten days from the year 2025 that are important to you, so you

can easily remember what day of the week they fell on. For example, one could choose days

such as the first day of school or the birthdays of different family members.

An example set of days important to me would be the following:

• January 1st (first day of the year): Wednesday

• March 12th (the day Real Madrid knocked Atlético out of the Champions League): Wednesday

• May 9th (my dad’s birthday): Friday

• June 10th (the day I finished my IGCSEs): Tuesday

• August 5th (my birthday): Tuesday

• September 4th (the first day of school): Thursday

• November 21st (my sister’s birthday): Friday

This set of days is just an example and everyone should choose dates that are important to

them and that they remember well.

(So for our example, we would remember that September 4th 2025 was a Thursday. That means that September 11th (7 days later) was also Thursday (because the week has 7 days so all Thursdays will be 7 days apart). This means that September 13th, which is 2 days later, was a Saturday (because Saturday is 2 days after Thursday).

2) Assign numbers to weekdays:

In this step, we are going to assign a number from 1-7 to each day of the week, following the order of the days in the week. This is such that:

• Monday = 1

• Tuesday = 2

• Wednesday = 3

• Thursday = 4

• Friday = 5

• Saturday = 6

• Sunday = 7

(Since we know that September 13th 2025 was a Saturday, we’ll keep the number 6 in mind)

3) Add the number of years ahead:

We now have to figure out how many years are between 2025 and the year we’re being asked about. If the year we’re being asked about is Y , we’d do the following calculation: Y - 2025. We would then add the result of this calculation to our original number obtained from step 2. (In our example, Y=2043, so we’d do the following calculation: 2043 - 2025 =18. We’d then add 18 to 6, the number we’d previously obtained in step 2, giving 24 (18+6).

4) Add leap years:

Up to now, I haven’t mentioned the most important difficulty: LEAP YEARS. We know that one takes place every 4 years. Therefore, the most recent one was in 2024. What we now have to do is calculate how many leap years there are between 2025 and the year of the date

we’re trying to calculate.

We will add this number to the number we had previously calculated by adding our numbers obtained in step 2 and step 3. It is also very important that the following rule is followed: if the day we are trying to find out about is on a leap year, we must only count that leap year if the day is after February 29th. Therefore, if the day is in January or February of a leap year, we don’t add 1 to our calculation. Furthermore, we must account for the fact that years that are multiples of 100 (except those that are multiples of 400) don’t have an extra day despite being divisible by 4.

(For our example, we know that the leap years between 2025 and 2043 are 2028, 2032, 2036 and 2040. That’s 4 leap years. We will now add 4 to 24, giving 28 (4+24).

5) Divide by 7 and find the remainder:

The final step is to divide the number from step 4 by 7 and find the remainder of this division. If the remainder is 1, the day we’re trying to calculate will be a Monday. If it’s 2, it will be a Tuesday. 3 gives a Wednesday. 4 gives a Thursday. 5 gives a Friday. 6 gives a Saturday. 0 gives a Sunday. Those are the 7 possibilities.

(In our example, we would divide 28 by 7. Since 28/7=4 with a remainder of 0, we know that

the day we’re trying to calculate -September 13th 2043- will be a Sunday.)

Once you understand this method, you can apply it repeatedly to future dates. It works because each normal year shifts the calendar forward by 1 weekday, while leap years shift it by 2. By adding these shifts and using modulus 7, we can quickly predict weekdays mentally. Ultimately, this method works reliably as long as you use the same reference date in the base year and then apply it to all the shifts across the years.

Of course, more advanced algorithms such as Zeller’s Congruence or John Conway’s Doomsday Algorithm can also be used for this purpose. Still, as a fast mental trick, this method is surprisingly effective — and it shows how mathematical patterns are often hidden in the most ordinary parts of everyday life.

Luis del Rivero (Year 12)