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?

Question

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.

Answer 1

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?

Answer 2

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.

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For completeness, though this has already been covered in another answer and comments:

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.