# How can I find the name of the last day of a week in certain year when the name of a day of a month for that year is given?

The problem is as follows:

Roger sees in his almanac that a certain month from a year the first day was monday and the last day was also monday. What day of the week, was the last day of that same year?

The alternatives given are:

$$\begin{array}{ll} 1.&\textrm{sunday}\\ 2.&\textrm{monday}\\ 3.&\textrm{tuesday}\\ 4.&\textrm{wednesday}\\ \end{array}$$

I'm lost at this question. The reason for it is that I don't know how many times in a year can the given condition be satisfied?.

Or would it happen just once a in a year?. How about leap years?. Will it be affected?. Can someone help me with this question?.

Since I'm a slow learner, an answer which could vastly help me the most is one which does include some graphic or visual aid to see how the days are "running" in the calendar or being arranged to fit as in this problem is indicating?. Can someone help me with that matter please?. I know that a straightforward answer could help but I require additional help to understand this please hence the necessity of a visual aid.

Let's assume that the first day of the month is a Monday. Then the next few Mondays of this month would be as follows: 8th, 15th, 22nd, 29th. For the last day to also be a Monday, we require a month that has $$29$$ days in it. Obviously this is a February during a leap year.

Of the $$14$$ possible calendar configurations, only one has a February starting on Monday and is a leap year. Can you use this to figure out the rest?

• I'm sorry I'm still stuck at this question. I added a paragraph in my question indicating that a visual aid would better help me to understand what's happening. Yes I got your point that the same day happens to be after seven days. But you assumed that it is february. But it could also happen in any month which has more than 29 days. However I think that the restriction impossed by the question is that the month ends on monday. Hence necesarily it has to be february. i.e let's say this year 2020?. But the part where I'm stuck is how to compute the last day for that leap year?. Mar 15, 2020 at 22:34
• I interpret the question to say that some month starts and ends on a Monday. So if it starts on Monday then we have the 1st of this month is a Monday. Adding $7$ consecutive times shows us the other days of the month that fall on a Monday. Likewise, for it to end on a Monday, one of these days (8,15,22,29) must be the last day of the month, and it's hopefully obvious which it is. Now for the last day of the year, note that there are $306$ days from February 29th to December 31st, and recall that $306 \equiv 5 \pmod 7$ Mar 15, 2020 at 22:51
• I'm counting the days, I obtained there are 306 days until 31st december that year and by dividing this to 7, I'm off by 5 days hence it should be saturday, which doesn't appear within the alternatives. But upon checking my calendar, 2016 had that configuration and the last day was saturday. My book says the answer is tuesday. Is it me, is it the book?, which is wrong?. Can you help me with that?. Mar 15, 2020 at 22:58
• I also got Saturday, so either I'm even more tired than I think or the book is wrong. Mar 15, 2020 at 22:59
• @WaveX A while ago I asked regarding the use of mod operators. I don't feel very comfortable working with them as I'm a novice in maths. But I suppose the intended meaning is that it is a multiple of 7 plus 5 is this the meaning?. I'm getting saturday. Am I wrong?. Mar 15, 2020 at 23:00

Here is a copy of the calendar from 2016, when February began and ended on a Monday. I made this image myself but you can check it against many other sources online.

The year ended on Saturday. Your calculation was correct and the book is wrong.

A month must have either $$28,$$ $$29,$$ $$30,$$ or $$31$$ days. Since each week has $$7$$ days, in order for the last day of the month to be on the same day of the week as the first day of the month, its day number must be $$1$$ plus a multiple of $$7.$$ Only $$29$$ fits both criteria.
So we're looking for a month with $$29$$ days. This can only be February. Counting the days in the ten months after February we find $$306$$ days, and since $$306 = 43 \times 7 + 5,$$ the remaining part of the year has $$43$$ weeks (which brings us to another Monday) plus five additional days, bringing us to Saturday. Or you can just refer to the calendar shown above.