This idea is not fully developed, I just wanted to put it up here and see what people think.

A cheshire cat dice has the special feature that whenever it lands on one side, that side disappears and you can't land on it again. That means that a cheshire cat dice can only be rolled six times, then it's used up. It also means that the more times it is used, the more predicable it gets.

Now, what would happen if we played some of the dice games here with cheshire cat dice? Backgammon? Dice Poker? Do you think it would change the strategy in an interesting direction? I imagine that the dice would be replaced every sixth move, and you would have some kind of special graphics to show which rolls are still possible.