(This post is actually a combination of 2 previous posts, so if it sounds familiar, that's why!)
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I don't know if I just have sneaky kids or if everybody has this problem, but I noticed several of my kids figured out pretty quickly that they could look at an open door at an angle and see where the balls are. For the 2nd day we played the game, I closed the door after they opened it and put a card on it so they would know which had already been chosen.
That's how I used my Cariboo game. I found my game at Goodwill last summer. It was originally priced for $5, but I told the worker that I wouldn't pay any more than $2 since it was missing some balls. If you don't have one, run to Kelly's and enter the giveaway!