If you restrict to months different from february, all of them have at least 30 days and you can easily see that "having" days 1..7 always gives you all the days up to 30 (just write numbers 1..30 putting a newline every 7 of them). Restricting to months from march to december (which always present the same pattern, whatever year you consider after the transition from Julian to Gregorian calendars) you can quickly check that all days 1..7 are there for all days of week (it suffices to see it for only one of them, then that's only a matter of modulo-7 maths).

So the only one that remains tricky is 31, because not all months "support" it.