My guess is that he has an auto reset Rubik's cube. Probably electronic or magnetic. Could he do that with a Rubik cube owned by a member of the audience? He had plenty of time to see the sides of the cube left on the desk and obviously he has skill with the cube also, so he also had time to copy the one in his hand to the one on the desk. That's what I think

That's a good show, JJ I think you're on to something, at least the reset function.

#1 These are 100% ordinary cubes (they said so in the show). Most of the permutations he uses look random but are solved by max 4 moves with one hand. The permutation at 3:42 has 3 checkerboarded sides and 3 uniform ones (U' D L2 U B2 D2 B2 D2 L F' U D' R U' F2 L F' B' for you nerds out there). It is all skillful manipulation. The matching cubes at the end are achieved by making Teller pick the cube Brundage handled at the beginning (he got lucky, otherwise he would have had to add some other elements to the routine), or by switching Penn's cube at 1:00 (looks suspicious). That way the pattern was something he could easily replicate.
That's how I think he did it. Pure skill and showmanship - no gimmicks.

I don't know how he did it. Amazing!

Did they found how he did it at the end of the show ?
That's the meaning of the show isn't it ?

#3 That's a boring, drawn out and over complicated theory. A much simpler one is auto reset, doesn't take months of wasted life practising what's lets face it is a pretty useless skill, time that could be spent doing something useful like working on a cure for cancer visiting an old person having sexual relations with a willing camel etc.

No resets, no switches just mathematics and memory using algorithms, a very impressive skill, slightly useless but great for parties!