I very recently became aware that Hanukkah 8th day sometimes (but rarely) falls on Tevet 3. What causes this to happen? At the same time, I also noticed that when Hanukkah 8th day/Tevet 3 happen together, this sometimes occurs in conjunction with the Gregorian New Year's Day (January 1). Additionally, when such triple coincidences occur, they always occur on the first day of the week (i.e. Sunday). One such "triple occurrence" happened most recently in 2017 (5777 Hebrew year). However, another one also occurred in 1797 (5557 Hebrew year), and yet another will apparently occur 150 years from now in the Gregorian year 2169 (5929 Hebrew year). So I now have a couple of related questions....
1) Can someone tell me the calendar conditions that cause Hanukkah 8th day to fall on Tevet 3?
2) Can anyone possibly extrapolate from that to tell me how frequently it might occur as a "triple event" with the Gregorian New Years Day? (I can only find two occurrences since the calendar shift from Julian to Gregorian happened in 1582). Thank you for all of your help.
BONUS QUESTION: Hebcal seems to indicate that another triple occurrence also happened in the Gregorian year 1550. I highly doubt that this is correct because this date is prior to adoption of the Gregorian calendar when the all important "11 day date shift" occurred - that would make the triple occurrence in Gregorian year 1550 theoretically impossible. So, it doesn't seem to be correct. Is it possible that Hebcal is not calculating correctly? Can someone check this out? Does anyone know how reliable Hebcal is?
Edit/Update: I became more curious, so I manually checked the entire date range from Gregorian years 1797 to 3797 for any "Hanukkah 8th day/Tevet 3/January 1" triple occurrences. This is what I found:
I found a mere eight triple occurrences of "Hanukkah 8th day/Tevet 3/January 1" over the entire span of 2000 years - 1797, 2017, 2169, 2275, 2389, 2609, 3486, and 3706. These 8 triple occurrences seemed to happen LESS frequently in future years according to Hebcal with a large gap between 2609 and 3486. The dates in Bold all occurred on Sunday. The dates in Italics all occurred on Friday. Furthermore, several of the dates appear to be equally spaced by 220 years (except for 2017 to 2169 and 2609 to 3486). I did observe that the Hebrew dates do seem to be shifting gradually (slightly) relative to the Gregorian solar calendar. I do not know the reason for this. I expected them to be continuous and relatively stable and in-sync. I also observed that starting with the Gregorian year 3032, Hanukkah/Kislev 25 begins to fall on January 1 (after that such "New Years Day/Kislev 25" occurrences happen regularly).
I suppose I may have missed one or two occurrences, but probably not many. I wanted to post this update for any who might be interested in the date patterns. Thank you for your answers.