MATH 136 DISCRETE MATHEMATICS
An introduction to the basic techniques and methods used in combinatorial problem-solving. Includes basic counting principles, induction, logic, recurrence relations, and graph theory. Every semester.
Required for a major or minor in Mathematics and in
Computer Science.
I teach it as a project-driven course in combinatorics & number theory. Taught to 74 students, 3 sections, in
2004–05. More than 1 in 6 Macalester students take this course.

“
Let us teach guessing
” MAA video, George Pólya,
1965
Points:
•Difference between wild and educated guesses
•Importance of testing guesses
•Role of simpler problems
•Illustration of how instructive it can be to discover that you have made an incorrect guess

“
Let us teach guessing
” MAA video, George Pólya,
1965
Points:
•Difference between wild and educated guesses
•Importance of testing guesses
•Role of simpler problems
•Illustration of how instructive it can be to discover that you have made an incorrect guess Preparation:
•Some familiarity with proof by induction
•Review of binomial coefficients

Problem: How many regions are formed by 5 planes in space?
Start with wild guesses: 10, 25, 32, …

random
Problem: How many regions are formed by 5 planes in space?
Start with wild guesses: 10, 25, 32, …

random
Problem: How many regions are formed by 5 planes in space?
Start with wild guesses: 10, 25, 32, …
Simpler problem:
0 planes: 1 region
1 plane: 2 regions
2 planes: 4 regions
3 planes: 8 regions
4 planes: ???

random
Problem: How many regions are formed by 5 planes in space?
Start with wild guesses: 10, 25, 32, …
Simpler problem:
0 planes: 1 region
1 plane: 2 regions
2 planes: 4 regions
3 planes: 8 regions
4 planes: ???
Educated guess for 4 planes:
16 regions

TEST YOUR GUESS
Work with simpler problem: regions formed by lines on a plane:
0 lines: 1 region
1 line: 2 regions
2 lines: 4 regions
3 lines: ???

TEST YOUR GUESS
Work with simpler problem: regions formed by lines on a plane:
0 lines: 1 region
1 line: 2 regions
2 lines: 4 regions
3 lines: ???
2
6
1
7
3
5
4

n
0
1
2
3
4
5
START WITH SIMPLEST CASE
USE INDUCTIVE REASONING TO BUILD
Space cut by n planes
1
2
4
8
Plane cut by n lines
1
2
4
7
Line cut by n points
1
2
3
4
5
6

n
0
1
2
3
4
5
START WITH SIMPLEST CASE
USE INDUCTIVE REASONING TO BUILD
Space cut by n planes
1
2
4
8
Plane cut by n lines
1
2
4
7
11
Test your guess
Line cut by n points
1
2
3
4
5
6

n
0
1
2
3
4
5
START WITH SIMPLEST CASE
USE INDUCTIVE REASONING TO BUILD
Space cut by n planes
1
2
4
8
15
Test your guess
Plane cut by n lines
1
2
4
7
11
Line cut by n points
1
2
3
4
5
6

GUESS A FORMULA n points on a line lines on a plane planes in space
0 1 1 1
1 2 2 2
2 3 4 4
3 4 7 8
4 5 11 15
5 6 16 26
6 7 22 42

GUESS A FORMULA n points on a line lines on a plane planes in space
0 1 1 1
1 2 2 2
2 3 4 4
3 4 7 8
4 5 11 15
5 6 16 26
6 7 22 42
 n

 k
 n k 0 1 2 3 4 5 6
0 1 0 0 0 0 0 0
1 1 1 0 0 0 0 0
2 1 2 1 0 0 0 0
3 1 3 3 1 0 0 0
4 1 4 6 4 1 0 0
5 1 5 10 10 5 1 0
6 1 6 15 20 15 6 1

GUESS A FORMULA n k–1 -dimensional hyperplanes in k -dimensional space cut it into:

 n

0



 n

1



 n

2


L


 n
 k

regions.

GUESS A FORMULA n k–1 -dimensional hyperplanes in k -dimensional space cut it into:

 n

0



 n

1



 n

2


L


 n
 k

regions.
Now prove it!

GUESS A FORMULA n k–1 -dimensional hyperplanes in k -dimensional space cut it into:

 n

0



 n

1



 n

2


L


 n
 k

regions.
Now prove it!
Show that if
 

# of regions with n hyperplanes in k -dimensional space, then
R ( n , k )

R ( n

1, k )

R ( n

1, k

1).
What do you have to assume about k

1-hyperplanes in k -dimensional space?

This PowerPoint presentation and the
Project Description are available at www.macalester.edu/~bressoud/talks