91av

Answer: 2036            

The winner is Carolyn Shakerchi of Surrey, UK.

Worked solution

There are 14 different calendars.

Seven that have 365 days each starting on a different day.

And there are seven that have 366 days each starting on a different day.

In eight years time in 2018 Joe will have collected all seven 365 day calendars, plus two 366 day calendars for the years 2012 & 2016.

The next 5 leap years are 2020, 2024, 2028, 2032 & 2036.

Each of the seven leap years will start on a different day as the frequency of leap years, one every 4 years, has no common factor with 7.

So in 2036 Joe will have a full set of calendars.