Simple question about the phases of the Moon If on March 20 the Moon were located at the position of the sun, how can I determine the phase in September 22, June 22, and December 20?

I drew the orbit and calculated by dividing sidereal days to get the position of moon. Then I moved the Earth to determine the position. But the answer was Full moon, First quarter, Third quarter respectively, which was different from mine.  I can’t figure out why I’m wrong.
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.