Suppose today was $14.04.2016$ (Thursday). Which weekday is on $14.04.2018$ and $14.04.2021$?

Question

How can one mathematically show which week of the day is on $14.04.2018$ and $14.04.2021$ if today was $14.04.2016$ (Thursday)?

I did the following: We know that $2016$ and $2020$ are leap years.

$2016$ had $366$ days.

Starting from $14.04.2018$ we can say that $366 - (31 + 29 + 31 + 14) = 261$ days left till 2017.

Now we add $1$ year ($365$ days) + $104$ (not $105$, because $2018$ is no leap year) + $261$ = $730$.

We take $730$ mod $7$ and get $2$.

So we add Thursday + $2$ days and get Saturday.

I'd do the same for $14.04.2021$.

Is this approach O.K.? Are there faster/better ways to do that (because I don't know how many days each month has by heart)?

Answer 1

Between 14.04.2016 and 14.04.2018 you have exactly two years with 365+365 days between them (no extra day in February in 2017. and 2018.). In total you have 730 days which is equal to 2 modulo 7. So you have two add two days to Thursday and the result is Saturday.

You don't even have to calculate the total number of days. It's useful to remember that 365 is equalt to 1 modulo 7. So if you have a two year span, the result modulo 7 would be $2\times 1=2$. Just be careful in case of a leap year - such year has 366 days which is 2 modulo 7.

Try to repeat the same exercise for 2021, just don't forget to add one extra day for leap year 2020.

Answer 2

Your method is fine. But: We know that 365 is 52 weeks and a day, meaning that each year the same date is a day ahead. So Wednesday 31/10/18 will be Thursday 21/10/2019. For a leap year it will be 2 days ahead instead. Using this we can simply add a day for each year and two days for each leap year. We also know that 2016's leap year has passed so only leap year to worry about is 2020.