# Mastermind

Players: 2
Ages: 8 and up
Cost: ~\$15 (Buy Mastermind on Amazon)
Math Ideas: Deductive reasoning, Permutations and combinations
Which colors are definitely not in the code?
What do these two guesses tell you, if you compare them?
How many possible codes are there in this game?

My wife really shouldn't let me go to Target. Every time, she sends me to get one simple item, and every time I show up at home with a new board game. I keep telling her "It's helping the kids learn!" but honestly they'd probably learn plenty from the other 28 board games we have...

But you, dear reader, get to benefit from my game addiction! A few weeks ago, I spied a game in retro packaging that I'd never seen before. I eagerly snapped it up and brought it home.

Within minutes of playing the game, I couldn't wait to see what my own students could do with the mathematical implications of the game. It's a great game for a 3rd or 4th grader, but the mathematical ideas that the game uses are taught in high school and college courses on discrete mathematics. Not that your kid will know that! To them, they're just guessing which colors match your secret code.

## How to Play

Mastermind is a codebreaking game that is similar to the game show Lingo, where people guess five-letter words and are told whether their letters are included in the secret word. If you imagine a game of Lingo that uses colors instead of words, you're pretty close.

To start a round, one person (the code maker) makes a code out of four colored pegs that they keep hidden from their opponent. Then the other player makes a guess, more or less at random, as to what the code might be. The code maker looks at the guess and compares it to the secret code. He then places his own pegs as follows:

• A black peg for any color peg that is the correct color and in the correct position
• A white peg for any color peg that is the correct color, but in the wrong position

The codebreaker then gets another chance to guess the code and gain even more information about the colors and locations of the color pegs. If the codebreaker guesses the code correctly within ten rounds, he wins!

The game was definitely too tough for my 5-year-old, but fortunately there is a version, Mastermind for Kids, that decreases the number of pegs and changes a couple of rules to make the game more accessible to younger kids.

## Where's the Math?

Once again, Mastermind is a game that contains no explicit numbers, yet is positively brimming with mathematical ideas. To begin with, the code maker has six different colors he can choose from for each part of his code. As a result, he has 6*6*6*6, or 1296 possible codes that he can create. Yet an adept codebreaker can narrow those 1296 possibilities down to a single code in only 10 guesses! How is that possible?

Let's look at an example: Imagine that your first guess, all blues, is a total whiff. You get no black pegs and no white pegs, indicating that there are no blues anywhere in the code. Suddenly, you realize there are only five options for each slot! You've narrowed your options down to 5*5*5*5, or 625. You've eliminated half the possible codes, simply by eliminating blue as an option. Not bad for a total bust of a guess!

Of course, I wouldn't expect your 3rd grader to get our a paper and start calculating all the possible combinations. Instead, I bet they would use less formal, but still deeply mathematical, forms of deductive reasoning to narrow in on the correct code.

Perhaps they would guess sets of four random colors until they stumble upon a guess that resulted in a black peg. One of their pegs is in the right spot! But which one? They could keep the leftmost peg the same and switch out all the rest, using trial and error until the determine which pegs are in the right spot.

I love Mastermind precisely because it lends itself so easily to having conversations between guesses. I'd suggest asking your child a question between every round!

The best question to ask, in my opinion, is "What did you learn from that guess?" The question is open-ended enough that you aren't guiding your child to a particular conclusion, but it keeps your child focused on the fact that she can learn something from every guess, even one that resulted in no pegs (like our all-blue example above).

Sometimes, kids notice something, only to forget that bit of information as they follow another path. I think it can be fruitful to ask your child to go back and revisit earlier guesses, asking "what does your new guess tell you, when you compare it to your old guess?"

This game, in many ways, is about building a case one bit of information at a time. The previous guesses are like the pieces of evidence that Sherlock Holmes uses to draw his remarkable conclusions. I recommend any questions that help your child make connections between the rounds of the game.

Mastermind is a game that is so rich in mathematical ideas, I might have to revisit it with more recommendations once I spend a few days analyzing it with my students next year! I think it could be the foundation of a wonderful statistics and probability unit for any middle school teacher.