# Mastermind: What should I guess next? That depends

Bob was playing a popular adventure game on his computer, and he came across a Mastermind minigame. Here's the results of his first four guesses:

Bob has never been great at solving logic puzzles on the spot, so he asked Alice what he should guess next. Alice is exceptionally good at logic puzzles, but has zero knowledge of video games or the context that this appeared in. Her response was "Well, that depends. If [...], then do this next: [...]. Otherwise, do this next: [...]"

What are the two different guessing strategies that Alice gave to Bob, and what condition did she give him to decide between them with?

Notes:

• This is a completely standard game of Mastermind within the video game. There are no special rules.
• Black dots mean "right color, right place" and white dots mean "right color, wrong place".
• The six colors that Bob already used in guesses are the only colors available.
• This is not a lateral-thinking puzzle.

I came up with this puzzle myself.

Where can the second clue get a black dot from? Not one of the B’s otherwise clue 1 or 3 is broken. Therefore the first letter is E. The fourth clue tells us the second and third letters are D and C, respectively. That only leaves three choices: EDCC, EDCD, EDCE.

Now the interesting part:

The obvious strategy is guess randomly, e.g. choose EDCC first, followed by EDCD and EDCE, but this risks requiring three more guesses to win. Alternatively we can guess CCAD next – which we know to be wrong. But the number of black/white dots will tell us what to do next with 100% certainty.

Therefore

Alice recommends the first strategy if Bob wants to maximise winning chances at every guess or the second strategy if Bob wants to guarantee winning within the next two moves.

Nice twist on the usual Mastermind logic and should be “gettable” for a solver who is switched on. Great puzzle!

• Nice answer! I think the first condition should be whether the game allows making a guess that is inconsistent with the previous result or not. Commented Aug 2, 2021 at 3:19
• @justhalf But it's already known that the game allows making a guess that is inconsistent with the previous result. If it didn't, then none of the second through fourth guesses would have been possible to make. Also, such a restriction would be a special rule, since standard Mastermind has no such rule. Commented Aug 2, 2021 at 16:25
• Ah ok, thanks for clarifying. I guess then the explanation can be made better a bit. Because currently option 1 has winning chance of {33%, 50%, 100%} and option 2 {0%, 100%}. I wouldn't say that option 1 is about maximizing winning chance at every guess, so it would be better to find a more apt description for it to distinguish the two options in a more logical way. What's your intended description for the two options? Commented Aug 3, 2021 at 1:06
• @justhalf That you can either guarantee a win in exactly two more guesses, or take a risk that might let you win in one, but might also make it take three. Commented Aug 6, 2021 at 15:47
• That's more like it. Commented Aug 7, 2021 at 7:15