# Why do Kisidran years only occur when starting on a Tuesday or non-leap Thursday?

A careful analysis of the Tur's calendar chart will reveal that the only time the months alternate throughout the entirety of the year (with the exception of Adar Aleph) is if Rosh HaShanah is a Tuesday, or if Rosh HaShanah is a Thursday in a non-leap year. In the remaining eleven year types, either Cheshvan will be 30 days or Kislev will be 29. Why is this?

I am aware that Rosh HaShanah can only be a Tuesday if the molad fell out between Chatzos HaYom on Monday and Chatzos HaYom on Tuesday (molad zakein), or the rare fourth dechiya, and that Rosh HaShanah can only be a Thursday if the molad fell out between Chatzos on Tuesday and Chatzos on Thursday (molad zakein and lo adu rosh). I'm also aware that the fluctuating lengths of Cheshvan and Kislev are in order to keep the Arba Dechiyos in check. (Source: Shvili d'Rakia, sections יא-יב.)

Modular arithmetic time:

385 mod 7 = 0
384 mod 7 = 6
383 mod 7 = 5
355 mod 7 = 5
354 mod 7 = 4
353 mod 7 = 3

That's how far Rosh Hashana moves from year to year. The Kisidran options are 4 and 6. So:

• if RH is on Monday, the next one would be on Friday or Sunday, which can't happen.

• if RH is on Tuesday, the next one would be on Saturday or Monday, which is fine.

• if RH is on Thursday, the next one would be on Monday or Wednesday, the latter of which is a problem.

• if RH is on Saturday, the next one would be on Wednesday or Friday, which can't happen.

So the only options for Kisidran years are Tuesday-Regular, Tuesday-Leap, or Thurday-Regular.