Discrete Mathematics Project

Counting Techniques Activity


Mastermind (Dan Snook)


1. Students will explore various counting methods.

2. Students will generalize solutions to a given problem.


This activity asks students to develop a plan to win at the game of "Mastermind". Students will need to use logic and reasoning to develop winning strategies. I first came across this activity in 1981 in a Combinatorics class using the text Applied Combinatorics by Alan Tucker.

Problem Statement

Discuss with your students that they have often played games of chance where their odds of winning come down to a draw of a card or roll of a die. It is important for them to realize that in many games follow mathematical principles, which applied correctly, can lead to methods of victory.

Instructor Suggestions

1. This activity is intended to last 2 days. Begin the first lesson with an explanation of the game, and then have students play the game for 30 minutes until they feel comfortable as to the rules.

2. The activity presents the students with some scored games, and some counting questions. Allow the students to work in groups on this part of the activity.

3. To wrap up the first day of class, simplify the game to 4 colors with no repetition. As homework, have the students develop an algorithm to find the secret code with the fewest possible guesses. (This is not easy, but it is possible that your 4th guess will always yield a score of XXXX.)

4. On the second day, have students present their methods by actually playing the game against the teacher. This is a great way of seeing if their method holds up!

5. Discuss why or why not certain methods work. Also, discuss the complexity of an algorithm that would solve the original game.


"Mastermind" worksheets, several Mastermind games, dry erase board


Class Period 1 Class Period 2
Introduction of the activity (5 minutes) Discussion of homework (full period)
Playing the game (20 minutes)  
Questions 2-6 (20 minutes)  
Assign Question 7 as homework  

Mathematics Concepts

Discrete Mathematics Concepts

Algorithms, Counting and Probability, Game Theory, Permutations

NCTM Standards Addressed

Problem Solving, Reasoning, Communication, Connections, Discrete Math

Colorado Model Content Standards Addressed

Number Sense (1), Data Collection and Analysis (3), Problem Solving Techniques (5), Linking Concepts and Procedures (6)

Curriculum Integration

This activity could be used in any high school math class. I would implement it in a Pre-Algebra class while we are working with probabilities and counting problems.

Further Investigation

After students have developed a winning strategy for 4 colors with no repetition, the more difficult task of generating an algorithm for the original game can be attempted.


The initial problem of selecting from 6 colors with repetition may seem overwhelming to some students. It may be more reasonable to change the initial set up to only four color choices and

not allow repetition.


Colorado Model Mathematics Standards Task Force. (1995) Colorado model content standards

for mathematics.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for

school mathematics. Reston, VA: Author:

Tucker, Alan (1980) Applied Combinatorics. New York: John Wiley & Sons

Last updated January 15, 1997