 Advanced gumball tip - optimal last digit resolution
Here's one additional refinement on Hamlet's advanced work. I had developed an essentially equivalent strategy independently. However, I find that I occasionally have narrowed it down to a single unknown digit, but that digit could be one of several remaining possibilities. If you arrive at this point and have MORE THAN TWO possible solutions, you can test for multiple numbers at once (up to three), instead one at a time. This technique guarantees you can find the missing digit with fewer steps than the number of possible digits.
Lets say you know the string is 1 5 2 ? 7, but the unknown could 8, 9, or 0. Forget about the ones you know, and enter '8 8 8 9 8'. No matter what the answer really is, you will find out with the game's first response. If it is a red ball, the unknown is 8. If it is a green ball, the answer must be 9. And if no ball is given, then it must be a 0. The three-number test is the best case (if you have four possibilities and get no ball, you can't know yet).
You can see that 1/3 of the time this will cost you one extra turn (if the answer was 8 in this example, your first guess would have been fine). But it will never cost you three turns, which is what one-by-one could take. That turn may be the one you don't have! There will still be cases where you're forced to guess, but this method will make a guess on your last chance necessary just a little less often. It also works with suspected duplicates, but that's even deeper water....
Apologies to Hamlet if you've already mentioned this - I didn't see it in your tutorial(s).
Last edited by williamson_39; 11-28-2007 at 10:16 PM.. | 
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