Professor John Mason of the Open University alerted me to the fact that the 3-color Tower of Hanoi puzzle is solvable if the goal is to affect a derangement of stacks of disks of the same color. The only restriction for a move is that a bigger disk can't be placed on top of a smaller one.
(To work out the puzzle, drag the disks from one peg to another. Drop it there.)
The puzzle is indeed solvable. You should not try it, though, if your patience is lacking. Time permitting I'll add an animated solution.
Stuart Anderson came up with a recursive algorithm for automatic solution of the puzzle. You can see how it works either Slow or Fast