Mastermind is a game of logic and reasoning. The objective of the game is to guess a secret code consisting of colored pegs. The secret code is a row of four pegs that may be chosen (with repeats) from the colors red, white, green, blue, yellow, and black. Each guess of the possible secret code is scored to give some information about how close the guess is to the secret code. Specifically, the player who chose the secret code indicates (1) how many of the pegs are of the correct color and in the correct position, and (2) how many of the pegs are of the correct color but in the incorrect position. This information is given in the form of a black marker, or X for (1) and a white marker , or O for (2). The game ends when all positions and colors have been correctly identified, resulting in four black markers. As we will see, this game has several interesting counting problems associated with it.

1. For practice, work in teams of two and try a few practices games to get an idea of how the game works.

2. Now, determine the secret code for the following guesses.

a)

Guess	Positions				Scoring
1	Green	Yellow	Red	Blue	O O O O
2	Green	Blue	Red	Yellow	O O X X

Secret Code:

b)

Guess	Positions				Scoring
1	Blue	Blue	Black	White	O X X
2	Yellow	White	Black	White	X X
3	White	Blue	Red	Green	O X
4	Yellow	Blue	Black	Red	X X

Secret Code:

3. Find a fourth guess whose scoring will allow you to determine the secret code for:

Guess	Positions				Scoring
1	Green	Yellow	Black	Red	X X
2	Yellow	Blue	Green	White	O X
3	Blue	White	Yellow	Yellow	O
4?

Explain why this guess will lead you to the correct solution.

4. Find the probability that your initial guess in a Mastermind game is correct.

5. What set of 4 or less X's and O's can never occur as Mastermind scores? Explain why.

6. How many different sets of X and O scores are possible ?

7. Consider the simplified game consisting of four pegs each of a different color. (Therefore, there is no repeating. How many secret codes exist now? Develop a complete strategy for playing this game so that you can determine the secret code in the fewest number of guesses.