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

The Cake-Cutting Problem Cupcake #2: Cherries Vanilla frosting A: 0  A: ½  X: ¼  X: 0  Chocolate A: ½  frosting X: ¾ 

The Cake-Cutting Problem How can they divide the cupcake fairly ? Example: Cherries Vanilla frosting A: 0  A: ½  X: ¼  X: 0  Chocolate A: ½  frosting 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 A: 0  A: ½  X: ¼  X: 0  Chocolate A: ½  frosting 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)

Cities and roads as a graph Example Fulton Albany Geneva Homer Cortland Ithaca Dryden Binghamton Elmira NYC

Cities and roads as a graph Example F 15 A 30 30 5 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” Example: Find the best path from G to N F 15 A 30 30 5 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” Example: Find the shortest path from G to N F 15 A 30 30 5 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” Example: Find the shortest path from G to N F 15 A 30 30 5 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” Example: Find the shortest path from G to N F 15 A Total length 30 = ? 30 5 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” Example: Find the shortest path from G to N F 15 A Total length 30 = 20 + 5 + 5 + 50 30 5 = 80 G 15 H 30 5 20 80 C 10 5 I 70 60 15 5 20 D 30 20 B 50 E 60 N

“The shortest - path problem” 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 F 1 A 0 1 5 G 0 H 3 0 5 2 C 1 2 6 I 3 0 5 1 D 0 2 B 2 E 1 N

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 13 th • 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!