Mastermind!

In the version of Mastermind I play (Iphone), you attempt to guess the color and location of 5 marbles from 9 possible marble colors (there can be more than one marble of the same color, btw). After you make your guess, feedback is provided in the form of a black dot for each marble of the correct color in the correct location, and a white dot for a marble of the correct color in the wrong location. My version of the game gives you 10 guesses before you lose.

I have two questions. Which feedback possibility (i.e, a single white dot, two white dots, etc) gives you the LEAST amount of information about the correct color/location of the marbles. The wrinkle here, of course, is that the absence of dots can actually give you a great deal of information; in the extreme example, feedback of zero black or white dots allows you to rule out 5 of 9 colors, putting you well on your way to solving the puzzle quickly.

The more general question is how would one mathematize the amount of information for all possible feedback possibilities? My brain just doesn't work well in this dimension.

Lastly a general comment to the Wizard: couldn't there be a cool casino game based on Mastermind? The little game I play is that if I can solve 3 puzzles in 20 guesses or less, I "win." Of course there's no math behind that threshold, it's just through experience that that seems sufficiently challenging that I can only do it about 1 out of 4 times.

A looooong time ago my brothers, cousins and I played that game nonstop, mostly by some beach or pool while on vacation. Must have been over 30 years ago. At one point we had a deluxe version. Wooden case/play board, glass marbles and metal pins. We also played lots of backgammon.

Misspent youth. We should ahve been playing BJ, poker and craps!

Anyway, there is a casino backgammon game (search for the thread). Real backgammon is a game of skill, with random factors thrown in. You can lose if you get bad throws of the dice, but you can also win despite bad throws. Casino backgammon is purely a game of chance.

Any casino mastermind would also be a game of chance. perhaps you'd draw cards or throw dice to "guess" ball color and position.

How about casino monopoly instead? Same game, same rules, only the bank (ie the house) charges a 5% commision on al transactions, and of course the money would be real.

The question im more interested in is whats the minimum amount of moves you could solve any puzzle in, assuming your following the same flow each time. EG start with colours 1-5, then based on feedback, change to next bit. basicly, whats the "basic strategy" and what, using that basic strategy would be the longest game? im assuming mastermind like this is solved... seems to have too few possibilities to not be solved...

Quote: Nareed

How about casino monopoly instead? Same game, same rules, only the bank (ie the house) charges a 5% commision on al transactions, and of course the money would be real.

I would totally do this! I would hang out and collect a few passes over "Go" take the couple hundred and split!

Quote: NicksGamingStuff

I would totally do this! I would hang out and collect a few passes over "Go" take the couple hundred and split!

Can you imagine the prop bets?

Quote: pingu

The question im more interested in is whats the minimum amount of moves you could solve any puzzle in, assuming your following the same flow each time. EG start with colours 1-5, then based on feedback, change to next bit. basicly, whats the "basic strategy" and what, using that basic strategy would be the longest game? im assuming mastermind like this is solved... seems to have too few possibilities to not be solved...

I haven't played any version of Mastermind in many, many years. However, back in the day, I remember reading that there definitely is a mathematical formula that can be used to solve any Mastermind game in a minimum number of moves. In those days, we played with 4 pegs, and if I remember correctly, it can be proven that 6 guesses will result in the correct answer.

Any Mastermind game will have a limited number of possibilities in the code. N^P. Where N is the number of colors, and P is the number of pegs. That fine number could then be used to figure out the minimum number of guesses that will result in the correct answer.

Obviously, any casino would be foolish to allow any game that could be won using mathematics alone. The only way it would work would be to determine the minimum number of guesses, and only allow winnings if the guesses take less than that. In other words, if 6 guesses in the Mastermind I played will result in a winning number, then only, maybe, 4 guesses or less could provide winnings. This would make it necessary for guessing, and lucky guessing at that, to win.

Quote: konceptum

The only way it would work would be to determine the minimum number of guesses, and only allow winnings if the guesses take less than that. In other words, if 6 guesses in the Mastermind I played will result in a winning number, then only, maybe, 4 guesses or less could provide winnings. This would make it necessary for guessing, and lucky guessing at that, to win.

This is along the lines of what I was thinking. The casino would design the game such that if a player plays with optimal strategy, the casino only has a slight edge (equivalent to a player-friendly blackjack house edge) - e.g., setting a "win/lose" line at X number of guesses. If a player plays sub-optimal strategy, the house edge increases (also like blackjack). But a player could get lucky and get it in fewer; the fewer guesses it takes, the more the payout. Jackpot if the player gets it on the first guess.

I like the idea of a monopoly game. How about some variant of a Yahtzee game? What game from your childhood would make a good casino table game?

Quote: ThreeCard

This is along the lines of what I was thinking. The casino would design the game such that if a player plays with optimal strategy, the casino only has a slight edge (equivalent to a player-friendly blackjack house edge) - e.g., setting a "win/lose" line at X number of guesses.

Would it be a table game or an electronic game? If table, how to randomize the choice of original setting of the color balls?

Anyway, the house could make a bundle on side bets. All balls the same color, four balls the same color, etc. A side bet could pay a bundle for guessing right on the first try, which would discourage good strategy.