A complex mathematical trick allows us to calculate the day of the week for any date. At first it will make us want to give up; but, if done well, this can really be a talent.
A modular arithmetic It has many applications, from encryption to magic tricks.
For example, what time is 5 o’clock after 10 pm? Let’s hope you say 3am and not 3pm. You know that 10 + 5 is not 3, but you also know that it must return to zero when it reaches 12. In this case, we divide 15 by 12 and keep the remainder. We write this as 10 + 5 = 3 modulo 12.
Through modular arithmetic it is possible to know the day of the week of any date – explains the mathematician Peter Rowlettthat .
As an example, let’s imagine that someone says January 4, 2025. To the number 4 we add 2, because this month started on a Wednesday, so we lost two days of the week at the beginning of the year. 4 + 2 = 6, so this date is the sixth day of the week, Saturday.
Later January dates require a little more work. For example, the 16th gives 16 + 2 = 18. There is no 18th day of the week, so we calculate the module 7which means we divide by 7 and use the remainder. 16 + 2 = 4 modulo 7, so January 16th is a Thursday.
We can calculate the day of the week for a date in 2025 by adding the day + 2 to the month number in the table below. Let’s choose a date at random: May 19th. The number of the month of May is 1, so we calculate 19 + 2 + 1 = 1 modulo 7so May 19, 2025 is a Monday.
| January | 0 | May | 1 | September | 5 |
| February | 3 | June | 4 | October | 0 |
| March | 3 | July | 6 | November | 3 |
| April | 6 | August | 2 | December | 5 |
But what does this table mean?
The trick is to decode the table of month numbers or calculate them quickly.
To help us, on March 31, 1887 (a Thursday), Lewis Carroll you posted a shortcut for these numbers: if a month starts or ends with a vowel, use 10 minus the month’s calendar number.
For example, for April, the fourth month, we use 10 – 4 = 6, so April’s month number is 6. It’s a fun coincidence that this works for April, June, August, and October.
We can get the number of the month following each of these by adding 2 if it is a 30-day month or by adding 3 if it is a 31-day month. So we just need to remember that January is 0, February and March are 3, and December is 5.
For dates in 2025, add 2 to the day. In 2026, 3 will be added, because 2026 starts on a Thursday.
It is standard shifts 1 day per year, or 2 in a leap year.
Carroll tried to simplify
Carroll’s solution was to work with the last two digits of the year: add to 6 the number of times that number divides by 12, the remainder and the number of times the remainder divides by 4.
For 2030, for example, we find that 30 can be divided by 12 twice, with a remainder of 6, which can be divided by 4 once. 6 + 2 + 6 + 1 = 15, or 1 module 7, so we add 1 to the dates of that year.
In a leap year, we deduct 1 for dates before March. Dates in the 1900s happened one day of the week later than in the 2000s, so we added 1 for years starting with 19.
Nobody said it was easy. But, if done well, you may have discovered a talent.
