Simple question about the phases of the Moon
The length of one lunar month is (on average) 29.53 days. You don't need to use sidereal days, since dates are based on a day of the sun, not of the stars. The calculation is just division

So you just need to count the number of days between March 20 and June 22 etc.

There are 94 days between those dates. And 94/29.53 = 3.18 lunar months.  So the moon will be crescent, on June 22nd (about 2 days before first quarter).  The other dates can be calculated similarly.

As a rough rule of thumb (that ignores things like the 30-31 pattern) the date of the new moon is earlier by about one day each month. By this reckoning, we would expect new moon on about June 17, Sept 14 and Dec 11, with first quarter one week later, etc.