Prove that every year contains at least $4$ months and at most $5$ months with $5$ Sundays

Question

Prove that every year contains at least $4$ months and at most $5$ months with $5$ Sundays each.

Miklos Bona rates this question as "less difficult than average" while I am stuck on it although I am able to solve most of the "average" exercises in his book.

I tried as follows:

There are $52$ weeks and $1$ day in a year, hence at least $52$ Sundays and at most $53$ Sundays. Let us group these Sundays into blocks of $5$ Sundays in each group, so we find that we can find $10$ complete "$5-Sunday$" blocks. We want to show that there are at least $4$ and at most $5$ months containing one such block each.

I understand that somehow I have to use the fact that a month can have $5$ Sundays only if the last Sunday comes on date $29,30$ or $31$. But I am not able to proceed further. Please provide some hint only and NOT a complete solution.

EDIT: Based on the hints, I have realized a solution (turns out I was necessarily complicating matters):

It is enough to show that there cannot be $3$ months or $6$ months having $5$ Sundays. The other cases are similar.

First we show that there cannot be $\leq3$ months having $5$ Sundays. Let the contrary happen i.e. let $n$ be the no. of such months, with $0\leq n\leq 3$. We note that every month has at least $4$ Sundays (the key point here, which I had missed) and that there are at least $52$ and at most $53$ Sundays in a year.

As $n$ months have $5$ Sundays each, a total of $5n$ Sundays are blocked and hence there are at least $52-2n$ and at most $53-5n$ Sundays to be distributed among $12-n$ months. Each such month has exactly $4$ Sundays so total no. of Sundays in the year$=4(12-n)+5n=48+n$ with $0\leq n\leq 3$. Hence the maximum no. of Sundays is $48+3=51$ while there are $52$ Sundays in a year, at least, which is a contradiction. So, $n\geq4$.

Let $n\geq6$ then no. of Sundays blocked by these "special" months = $5n$. So total no. of Sundays in the year $=48+n$ with $n\geq6$ so total no. of Sundays is at least $54$, again a contradiction as total no. of Sundays is at most $53$. Hence $n\leq5$.

Answer 1

Every month contains $\geq28$ days hence at least $4$ sundays.

If the number of months containing $5$ sundays exceeds $5$ then what can be said about the total number of sundays in a year?

Answer 2

Going along with your train of thought,

HINT: how many Sundays must each month have?

HINT: distribute the Sundays (minimally) to each month using the previous hint. What happens to the rest of the Sundays?

This will have to use a variant of Pigeon-hole.

Answer 3

1. As you say, every year has 52 or 53 Sundays.
2. Every month has 4 or 5 Sundays.
3. Every year has exactly 12 months.

That is all the information you need to prove the assertion. You don't need to worry about 29's, 30's and 31's.