You can navigate the grid below from the top-left yellow circle to the bottom-right yellow square by making only orthogonal (not diagonal) moves, going from one position to the next only if the shape and/or the color of the shape in the entered cell is the same as that of the cell you are leaving.
What path has the minimum number of moves without going through the same cell twice?
What path has the maximum number of moves without going through the same cell twice?
(In reply to re: Best Minimum?
I think this was my original solution, although looking back on it there is an initial "D-" hanging on a line above the rest of the sequence that would be easy to miss -- guess I should have put it all on its own line. But nonetheless I think we have the best minimum so far, and as far as I can tell you have the best maximum (so far).