Well, I took Gary's advice and just got off the phone with Bose Tech support. Here is exactly what he said.

If you are running a Bose L1 Classic there will be no technical problem using just the bottom part of the tower because each part of the tower on the classic has it's own amp.

If on the other hand you have the L1 Mod1 or Mod II there will be an impedance mismatch when running only one part of the tower because on those systems a single amp feeds both tower parts.

(I thought it was of interest that the Mod 1 and Mod II risk damage due to the impedance mismatch of using only one tower piece.)

Concerning how many cables to run - if you are running only mono it doesn't matter. Nor does it matter which channel you run the cables into. If you are attempting to run stereo then use two cables.

These answers address only the electrical requirements of each system and do not address the audio components such as wave cancellation etc..

Hammer