It would be really nice to see a simple formula
Here is a formula that could be useful. Names of variables may or may not be unambiguous but try the maths.
Magnet pair pitch (physical angle)= 720 / number of magnets
coil pitch (physical angle)= 360 / number of coils
phase shift (phase angle)= 360*(Magnet pair pitch-coil pitch)/Magnet pair pitch
..... and then you work with that shift to see how many phases you get.
example. 8 magnets, 6 coils, magpitch = 90, coil pitch = 60, phase shift =360*(30/90) = 120 degrees
this gives a simple 3-phase output.
example 12 magnets, 10 coils, magpitch 60, coilpitch 36, phase shift = 360*(24/60) = 72 degrees
which gives a 5-phase output.
10 magnets, 9 coils, magpitch 72, coilpitch 40, phase shift = 360 * (32/72) = 160 degrees
which could be 9-phase.
However you can make a pretty workable 3-phase if you reverse some coils. Connect 1,-2 and 3 in series, 4,-5 and 6 in series and finally 7, -8 and 9 in series.
It helps to make a diagram of the phase angles and see how best to group them, never forgetting that you can reverse the wires to a coil and change the phase angle by 180 degrees.
all good fun