David M. Bressoud Macalester College, St. Paul, Minnesota MathFest, Albuquerque, NM

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David M. Bressoud Macalester College, St. Paul, Minnesota MathFest, Albuquerque, NM August 5, 2005

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 ” 1965 Points: MAA video, George Pólya, •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 ” 1965 Points: MAA video, George Pólya, •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

START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD

n

0 1 2 3 4 5 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

START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD

n

0 1 2 3 4 5 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

START WITH SIMPLEST CASE USE INDUCTIVE REASONING TO BUILD

n

0 1 2 3 4 5 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

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

GUESS A FORMULA

n

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

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