LS, according to your logic, I can put 175 in January alone and leave 177 in June.
To get the smallest possible number in June, one should try to put the number EVENLY among other months, not try to create some extra hurdles (such as creating a month with 31 people).
This is similar to voting in an election or gambling in Las Vegas: Stick to the same strategy ALL the time.
If you first spread 29 people evenly among 12 month, you should spread the extra 4 people EVENLY among the months AGAIN. Plain and simple. What you get is a few months with 30 people. Then the smallest possible number for June to become the most popular month is 31, just one extra than the rest.
Remember, your goal is to find the LOWEST possible number for June to win the contest. So do not pile up extra people in any month other than June. Spread them out!