It is impossible to put Z patterns on all six faces of the
cube, just as it is impossible to extend the Laughter (or Zig-Zag)
pattern to six faces. The problem is with the corners. If every
face is to have a pair of opposite corners that agree with the face
center and a pair of opposite corners that agree with each other
but not with the center, then the only constructible pattern is
like (i.e. M-conjugate to) the following:
F - U - U - U - FD - L F - R U - R B - L - L - - F - - R - - B - L - D R - F R - U L - BB - D - D - D - B
But this pattern has incorrect corner orientation, and so is not
achievable.
The T patterns were introduced to this list by David C.
Plummer [3 September 1980 2123-EDT], who assigns credit to Tanya
Sienko for the idea. Jim Saxe and I [27 January 1981 0102-EST] gave
a process for Tanya's T, the pattern that Minh Tai uses to sign
with. Our process, (FF UU)^3 (UU LR')^2, is four quarter-twists
shorter than Tai's ``quick routine'' because of the cancellation in
the middle. The other T pattern, Plummer's T, can be achieved in 28
qtw: FF UD' F'B' RR F'B U'D RL FF RL' UD' RL FF R'L U'D'.