Edges of Glory Glorious exploration in math and its applications in - - PowerPoint PPT Presentation

Edges of Glory

Glorious exploration in math and its applications in our daily life!

Part 1: “The Cake-Cutting Problem”

A Sprint data plan commercial

“Sharing is caring!”

Sometimes, we only have a limited amount of things, and we have to share! Example:

Things we want a lot of Things we don’t want a lot of Cell phone data plan Chores

Sharing “fairly”?

Example: Sharing cell phone data plan, based on

• Number of children?
• Amount of hair?
• Amount of dental work?
• …?

The Cake-Cutting Problem

Xiaoting and Alice have to share one delicious chocolate cupcake

The Cake-Cutting Problem

Xiaoting and Alice have to share one delicious chocolate cupcake How can they divide the cupcake fairly ? They both love chocolate cupcakes and want a piece that is as big as possible!

The Cake-Cutting Problem

Xiaoting and Alice have to share one delicious chocolate cupcake How can they divide the cupcake fairly ?  But, what do we mean by “fair”?

The Cake-Cutting Problem

A division of the cake is fair if

the value Alice assigns to her piece is equal to the value Xiaoting assigns to her piece (happiness)

Example:

A division that is fair!

The Cake-Cutting Problem

Xiaoting and Alice have to share cupcake #2.

The Cake-Cutting Problem

Cupcake #2:

Cherries Vanilla frosting Chocolate frosting

The Cake-Cutting Problem

Cupcake #2: Xiaoting and Alice like the parts differently

Alice Xiaoting Cherries Indifferent Like! Vanilla frosting Like! Indifferent Chocolate frosting Like! Really like!

The Cake-Cutting Problem

Cupcake #2: Xiaoting and Alice value the parts differently

Alice Xiaoting Cherries 0  ¼  Vanilla frosting ½  0  Chocolate frosting ½  ¾ 

The Cake-Cutting Problem

Cupcake #2:

Cherries Vanilla frosting Chocolate frosting A: 0  X: ¼  A: ½  X: 0  A: ½  X: ¾ 

The Cake-Cutting Problem

How can they divide the cupcake fairly ? Example:

Cherries Vanilla frosting Chocolate frosting A: 0  X: ¼  A: ½  X: 0  A: ½  X: ¾ 

The Cake-Cutting Problem

Xiaoting and Alice have to share cupcake #2. How should they divide the cupcake fairly ? Alice: How about I cut the cupcake into 2 pieces, then you choose the piece that you want? Xiaoting: Sounds great!

The Cake-Cutting Problem

An algorithm for cutting a cake fairly

Step 1: Alice cut the cupcake into 2 pieces (any size) Step 2: Xiaoting gets to choose the piece that she wants first Step 3: Alice gets the remaining piece

Cherries Vanilla frosting Chocolate frosting A: 0  X: ¼  A: ½  X: 0  A: ½  X: ¾ 

The Cake-Cutting Problem

How if we want to divide a cake fairly among three or more people? Form five groups!  Try to share the cake fairly among your group members.

The Cake-Cutting Problem

How if we want to divide a cake fairly among three or more people? This is a hard problem! One possible solution: the “moving knife solution”

The Cake-Cutting Problem

Conclusion!

• Sometimes, we have to share
• We want to share fairly, but doing this is

sometimes hard

• Math can help a group of people decide the

best way to share such that everyone gets their fair share

Part 2: Graphs

Driving directions in Google Maps

Cities and roads as a graph

A graph is a collection of

• Nodes (to represent cities or intersections)
• Edges that connect pairs of nodes

(to represent roads)

Homer

Cities and roads as a graph

Example

Elmira Dryden Albany Ithaca Geneva Fulton Cortland NYC Binghamton

Example

Cities and roads as a graph

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 5

Example: Find the best path from G to N

“The shortest-path problem”

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 5

Example: Find the shortest path from G to N

“The shortest-path problem”

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 5

Example: Find the shortest path from G to N

“The shortest-path problem”

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 5

Example: Find the shortest path from G to N

“The shortest-path problem”

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 Total length = ? 5

Example: Find the shortest path from G to N

“The shortest-path problem”

E D A G F C N B H I 30 30 20 10 15 30 60 20 50 5 5 30 15 15 80 5 70 60 20 Total length = 20 + 5 + 5 + 50 = 80 5

Example: Find the best path from G to N

• Shortest in distance
• Shortest in time
• …

“The shortest-path problem”

Example: Find the best path from G to N

• Shortest in distance
• Shortest in time
• Cheapest in toll fees
• Most scenic
• Passes by the most number of candy stores
• …

“The shortest-path problem”

Example: Find the best path from G to N

• Shortest in distance
• Shortest in time
• Cheapest in toll fees
• Most scenic
• Passes by the most number of candy stores
• …

“The shortest-path problem”

Example Find the path from G to N that passes through the most number of candy stores

“The shortest-path problem”

E D A G F C N B H I 3 5 1 1 2 2 2 5 1 1 2 5 6 3 1

What else are graphs good for?

• To visualize and to study social networks

What else are graphs good for?

• To visualize and to study social networks
• To model and help prevent spread of disease
• Many other interesting mathematical

problems!

• Another example…

Six Degrees of Kevin Bacon

• Social network of actors and actresses

– Edge if two people appear in a movie together

• Leonardo DiCaprio’s Kevin Bacon number is 2

– Worked with Tom Savini in Django Unchained – …who worked with Kevin Bacon in Friday the 13th

• Idea : Almost every actor has a Kevin Bacon

number smaller than 6

The Oracle of Bacon

Conclusion

• We used math to analyze two important real-

world problems

– Cake-cutting (Resource sharing) – Shortest path

• Math may appear to be a boring subject
• …but can be used to do glorious things!

Thank you!