You are tasked with reforming the Gregorian calendar to make

You are tasked with reforming the (Gregorian) calendar to make it easier to use. Instead of the twelve months having numbers of days that vary so much (from 28 to 31), you are going to even things up more (add days to February, etc.). Show that no matter what you do with twelve months, there must be some month with at least 31 days.

Solution

We have to show that some month should have atleast 31 days. Let us assume that a month has 30 days i.e there will be 12x30=360 days in a year. But a year has 365 days. To add these extra 5 days, either there should be more than 12 months or any month should have more than 30 days i.e atleast 31 days. In order to have 365 days in a year, having more than 12 months is not possible. So we have to go with the other option i.e any month should have more than 30 days. Therefore, this shows that there must be some month with at least 31 days.