Which day of the week will be $100$ days from now

Question

If today is Wednesday, what day of the week will it be $100$ days from now? Going forward $1,8,15,\dots,92,99$ days all result in Wednesday. So the $100$th day will be Thursday. But the solution given by PASCAL contest of University of Waterloo says Friday. Can somebody explain please?

Answer 1

Today is Wednesday

• $1$ day from now it is Thursday.
• $2$ days from now it is Friday.
• $3$ days from now it is Saturday.
• $4$ days from now it is Sunday.
• $5$ days from now it is Monday.
• $6$ days from now it is Tuesday.
• $7$ days from now it is Wednesday.
• $14 = 7\cdot 2$ days from now it is Wednesday.
• ...
• $70 = 7\cdot 10$ days from now it is Wednesday.
• ...
• $98 = 7\cdot 14$ days from now it is Wednesday.
• $99 = 7\cdot 14 + 1$ days from now it is Thursday.
• $100 = 7\cdot 14 + 2$ days from now it is Friday.

Answer 2

You have everything off by one. Wednesday will evidently be $7$ days from now, not $8$. Consequently, dividing $100$ by $7$ you have a remainder of $2$ (14 weeks and 2 days), and thus $100$ days from now will be two days after Wednesday, or Friday.

Answer 3

The only thing you need to know is some basic modular arithmetic.

We have $$100=7\cdot 14+2$$ or, in the other words, $$100\equiv 2 \pmod 7.$$

In any $100$ days period you have fourteen weeks and two extra days. The two extra days are the only thing which has influence on the end result. (Simply because waiting one week does not change the day of the week. The same is true for two weeks or any multiple of seven days.)

This means that if today is Wednesday, $100$ days later it will be Friday. (The answer must be the same as for the question: What day will it be two days after Wednesday?)

---

I was a bit surprised that Wolfram Alpha can answer the question even if I formulate it just as a question in English language: What day of the week will it be 100 days after Wednesday?

Of course, it is not surprising that mathematical software can do calendar computations. But they seem be rather by changing question entered in English language to a mathematical question. (Although you can see that WA interpreted it as: What day is 100 days after this Wednesday? Which is a bit different. But it is not difficult to realize that it does not matter whether we take one particular Wednesday or whether we ask the same question for any Wednesday.)

---

This question also appeared on BBC's University Challenge. See this article and you probably find some videos on youtube.

Answer 4

Step 1-The same day comes after 7 days.S0, 100/7= 98 = current day Step 2- add the remaining 2 days to the current day (100/2)+2 = required day.