Synchronous speed, n1 = (120 * fn)/p where p = poles
Synchronous speed, n1 = (60 * fn)/p where p = pole pairs
Most text book presentations perform calculations using poles, but some use pole pairs.
In an induction motor, the rotor poles are formed by the action of the stator magnetic field. The number of rotor poles must always match the number of stator poles. For most other types of motors, the rotor is constructed so that the number of rotor poles matches the number of stator poles.
In induction motors, the slip at the rated frequency and load is between about 1.5% and 3% of synchronous speed for "standard" motors. To calculate the number of poles, use the formula to find the synchronous speed that is about 1.5% to 3% higher than the loaded speed. For "high slip" motor designs, the synchronous speed might be as much as 15% above the loaded speed. A wound-rotor motor with external resistance could operate over a wider speed range, but should have a speed rating stated without external resistance. In that case, the loaded speed would probably be 3% to 5% below the synchronous speed.
Single-phase motors generally have higher slip at full load than three-phase motors, but most of them will have 3% to 5% rated slip. Shaded-pole motors may be a little higher.