Shows that for K number of matched single-coil pickups, the total number of possible in-phase pairs is K*(K-1)/2, and the total number of possible contra-phase pairs is K*(K-1)/2. The ultimate number of unique humbucking pairs for K is UK = K*(K-1) and the total number unique humbucking tones or timbres (due to series and parallel connections) is 2* UK = 2* K*(K-1). So for K = 2,3,4,5,…, UK = 2, 6, 12, 20, …, and the number of possible unique timbres is 2* UK = 4, 12, 24, 40, …
For K matched single-coil pickups, there are 2 to the (K-1) power (2^(K-1)) different unique sets of N and S pole arrangements, of which it is likely that K sets will produce 2* UK unique humbucking timbres.
So for K = 4, there are 8 different unique pole arrangements. These arrangements can go from all N (or all S) with 12 different contra-phase outputs, to 1 N and 3 S (or 1S and 3N) with 6 in-phase and 6 contra-phase outputs, to 2N and 2S with 8 in-phase and 4 contra-phase outputs.
If a guitar with 4 pickups in fixed positions, which can each be switched out for a pickup of the opposite pole, and the switching system can compensate to produce the K*(K-1) = 12 unique timbres (for each of the 8 pole sets) in increasing progression from warm to bright, then a 4-pickup guitar can have up to 8 different personalities.
If the pickups can be shifted in physical position, to shade the timbres, then a whole new universe of customization opens up.