I used a handy table by Jean Meeus ( 1995 page 405) in his wonderful reference book 'Astronomical Tables of the Sun, Moon and Planets' to grind out the simple estimate for when a second Full Moon happens on June 30th at 12:00 Universal Time. I computed all the Full moons that seem to be happening around this date and time, and then looked at the best candidates to calculate their phase at this date and time, I then determined their day of the week. This is what I got:
Year Day Phase 1901 Sunday 0.99 1912 Sunday 0.99 1920 Wednesday 0.99 1931 Tuesday 1.0 1939 Friday 0.98 1977 Thursday 0.99 1996 Sunday 1.00The 1901 and 1912 dates seem to be the best candidates. I did these in some detail and found that the 1901 Full Moon on June 30 was only 98 percent full so this was eliminated. The 1912 Full Moon was 99 percent Full at 12:00 UT on June 30 and was Declining from its Full phase earlier that day. To the best of my ability in the time I have, it looks like the 1912 Full Moon happened on June 30 which was a Sunday. If anyone has a good planetarium simulator, you might want to check this out for locations in North America. It may have been the only other Blue Moon in June on a Sunday this century!