Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon...
Transcript of Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon...
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Painting yourself into a corner:Graph colouring and optimization
Andrew D. King
Simon Fraser University, Burnaby, B.C.
A taste of π, December 1, 2012.
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A little about me
É Studied math and computer science at University ofVictoria, University of Toronto, and McGill (Montréal).
É Worked at Institut Teoretické Informatiky, Prague, CZ,Columbia University, New York, and Simon FraserUniversity, Burnaby.
É I am a postdoctoral researcher: I finished my Ph.D.recently, and I research unsolved math problems.
É My research area is kind of like really complicatedsudoku.
É We call it graph theory.
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A little about me
É Studied math and computer science at University ofVictoria, University of Toronto, and McGill (Montréal).
É Worked at Institut Teoretické Informatiky, Prague, CZ,Columbia University, New York, and Simon FraserUniversity, Burnaby.
É I am a postdoctoral researcher: I finished my Ph.D.recently, and I research unsolved math problems.
É My research area is kind of like really complicatedsudoku.
É We call it graph theory.
![Page 4: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/4.jpg)
A little about me
É Studied math and computer science at University ofVictoria, University of Toronto, and McGill (Montréal).
É Worked at Institut Teoretické Informatiky, Prague, CZ,Columbia University, New York, and Simon FraserUniversity, Burnaby.
É I am a postdoctoral researcher: I finished my Ph.D.recently, and I research unsolved math problems.
É My research area is kind of like really complicatedsudoku.
É We call it graph theory.
![Page 5: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/5.jpg)
A little about me
É Studied math and computer science at University ofVictoria, University of Toronto, and McGill (Montréal).
É Worked at Institut Teoretické Informatiky, Prague, CZ,Columbia University, New York, and Simon FraserUniversity, Burnaby.
É I am a postdoctoral researcher: I finished my Ph.D.recently, and I research unsolved math problems.
É My research area is kind of like really complicatedsudoku.
É We call it graph theory.
![Page 6: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/6.jpg)
A little about me
Jessica McDonald speaking at a conference
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A little about me
Daniel Kral figuring out some configurations in Montreal
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A little about me
This is what the configurations turned into. We solved anold problem... older than me!
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A little about me
My knowledge of Czech: pozor and nemazat. Useful!
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A little about me
Lots of good reasons to pozor.
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A little about me
This is what the configurations turned into. We solved anold problem... older than me!
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A little about me
The research institute in Prague (yes, really)!
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A little about me
A workshop in Barbados. Outdoor blackboards!
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A little about me
Working with my Ph.D. advisor and officemate
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A little about me
Pondering graph theory in Italy
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A little about me
I said I study graph theory.What’s a graph?
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What is a graph?
Graff: graffiti
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What is a graph?
Graf, Steffi
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What is a graph?
Graf, Iowa
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What is a graph?
GTPase Regulator Associated with Focal Adhesion Kinase(GRAF)
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What is a graph?
The graph of a function.
Close, but not that kind of graph.
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What is a graph?
The graph of a function.Close, but not that kind of graph.
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What is a graph?
A graph is a network of objects!
É Dots: vertices or nodes.É Lines: edges or connections.
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What is a graph?
A graph is a network of objects!
É Dots: vertices or nodes.É Lines: edges or connections.
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What is a graph?
A graph is a network of objects!
A social network, for example.The objects are people.
The connections represent relationships.
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What is a graph?
A graph is a network of objects!
A social network, for example.The objects are people.
The connections represent relationships.
![Page 27: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/27.jpg)
What is a graph?
A graph is a network of objects!
A social network, for example.The objects are people.
The connections represent relationships.
![Page 28: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/28.jpg)
What is a graph?A graph isÉ A set of dots called verticesÉ Some pairs of dots are connected.É The connections are called edges.É Two vertices are connected or adjacent if they have
an edge between them.
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What is a graph?A graph isÉ A set of dots called verticesÉ Some pairs of dots are connected.É The connections are called edges.É Two vertices are connected or adjacent if they have
an edge between them.
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What can vertices and edges represent?A graph can represent something abstract andmathematical or something practical... or both!
É Vertices represent combinations of two buttons.É Exercise: Draw an edge between each pair of
combinations that doesn’t share a button.
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What can vertices and edges represent?A graph can represent something abstract andmathematical or something practical... or both!
É Vertices represent combinations of two buttons.É Exercise: Draw an edge between each pair of
combinations that doesn’t share a button.
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What can vertices and edges represent?A graph can represent something abstract andmathematical or something practical... or both!
É Vertices represent tennis games between two players.É Exercise: Draw an edge between each pair of games
that can happen simultaneously.
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What can vertices and edges represent?A graph can represent something abstract andmathematical or something practical... or both!
É The resulting graph is the same either way.É Graphs provide a flexible way to model real-life
problems in a mathematical setting.
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What can vertices and edges represent?A graph can represent something abstract andmathematical or something practical... or both!
É The resulting graph is the same either way.É Graphs provide a flexible way to model real-life
problems in a mathematical setting.
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Erdos Numbers
Paul Erdos, 1913–1996: A true master of graph theory.
É Published over 1500 papersÉ N is a Number – A documentary about him on
Youtube.
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Erdos Numbers
Erdos numbers involve interconnectedness ofmathematicians!
É Take a graph where the vertices are mathematicians.É Two mathematicians are connected if they published
a paper together.É How long is the shortest path from me to Erdos?É That’s my Erdos number.
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Erdos Numbers
Erdos numbers involve interconnectedness ofmathematicians!
É Take a graph where the vertices are mathematicians.
É Two mathematicians are connected if they publisheda paper together.
É How long is the shortest path from me to Erdos?É That’s my Erdos number.
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Erdos Numbers
Erdos numbers involve interconnectedness ofmathematicians!
É Take a graph where the vertices are mathematicians.É Two mathematicians are connected if they published
a paper together.
É How long is the shortest path from me to Erdos?É That’s my Erdos number.
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Erdos Numbers
Erdos numbers involve interconnectedness ofmathematicians!
É Take a graph where the vertices are mathematicians.É Two mathematicians are connected if they published
a paper together.É How long is the shortest path from me to Erdos?
É That’s my Erdos number.
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Erdos Numbers
Erdos numbers involve interconnectedness ofmathematicians!
É Take a graph where the vertices are mathematicians.É Two mathematicians are connected if they published
a paper together.É How long is the shortest path from me to Erdos?É That’s my Erdos number.
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Colouring a mapI am a cartographer on a budget. I can print my map infour colours, but neighbouring states should get differentcolours. Is this always possible?
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Colouring a mapI am a cartographer on a budget. I can print my map infour colours, but neighbouring states should get differentcolours. Is this always possible?
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Colouring a mapI am a cartographer on a budget. I can print my map infour colours, but neighbouring states should get differentcolours. Is this always possible?
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Colouring a mapHow can we view this map as a graph? Remember,neighbouring states should get different colours.
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Colouring a mapHow can we view this map as a graph? Remember,neighbouring states should get different colours. Verticesare states. Neighbouring states are connected.
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Colouring a mapNow we want to give the vertices colours so that noadjacent (connected) vertices get the same colour!This is the graph colouring problem.
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Colouring a mapLook at Washington State.
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Colouring a mapLook at Washington State. Capital city?
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Colouring a mapLook at Washington State. Capital city? Olympia.
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Colouring a mapLook at Washington State. Capital city? Olympia.Washington only neighbours two states:
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Colouring a mapLook at Washington State. Capital city? Olympia.Washington only neighbours two states:Oregon and Idaho.
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Colouring a mapLook at Washington State. Capital city? Olympia.Washington only neighbours two states:Oregon and Idaho.
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Colouring a mapIf we can 4-colour everything except Washington, we canextend the colouring to Washington.At least 4-2=2 colours are available for WA.
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Colouring a mapWe can make the same argument for CA, FL, ME, NH, VM,RI, LA, GA, MI, NJ, CT, DE, ND, and MD.Oh and HI and AK, obviously.
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Colouring a mapSo we can ignore these states, colour the rest, thenextend the colouring when we are done.
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Colouring a mapBut we can repeat this argument. For example, Oregononly has two grey neighbours. We can remove a bunch ofstates this way!
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Colouring a mapBut we can repeat this argument. For example, Oregononly has two grey neighbours. We can remove a bunch ofstates this way!
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Colouring a mapNow we repeat the argument. Our map gets smaller...
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Colouring a mapNow we repeat the argument. Our map gets smaller...And smaller...
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Colouring a mapNow we repeat the argument. Our map gets smaller...And smaller...And smaller.
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Colouring a mapNow we can colour the map in stages. First colour thestate numbered 1, then extend to states numbered 2,then 3, etc.
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Colouring a mapExercise: Can you colour the states with 4 colours, stageby stage, so no two neighbouring states get the samecolour?
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A recursive 4-colouring algorithm
We just performed a recursive 4-colouring algorithm:É Recursive: To solve the problem on our graph, we
solve the same problem on a smaller graph.É 4-colouring: We colour the graph using 4 colours.É Algorithm: A step-by-step method for solving a
problem.
To 4-colour a map G:1. Find a vertex v with at most 3 neighbours (e.g. WA).2. Remove v and recursively 4-colour what remains.3. Since v has at most 3 neighbours, we can extend the
colouring to v.Does this always work?
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A recursive 4-colouring algorithmTo 4-colour a map G:1. Find a vertex v with at most 3 neighbours (e.g. WA).2. Remove v and recursively 4-colour what remains.3. Since v has at most 3 neighbours, we can extend the
colouring to v.Does this always work?
Absolutely not! Maybe v does not exist. :(
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A recursive 4-colouring algorithmTo 4-colour a map G:1. Find a vertex v with at most 3 neighbours (e.g. WA).2. Remove v and recursively 4-colour what remains.3. Since v has at most 3 neighbours, we can extend the
colouring to v.Does this always work?
Absolutely not! Maybe v does not exist. :(
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Are 4 colours always enough?
The graph associated with a map is called a planar graph,because it can be drawn in the plane (2-dimensionalspace) without any edges crossing each other.
Question (1852):Is every planar graph 4-colourable?
Answer (Appel and Haken, 1976):Yes!
Answer (Robertson Sanders Seymour Thomas, ’95):Yes!
Answer (Werner and Gonthier, 2005):Yes!Why did they prove the Four Colour Theorem so manytimes?
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Are 4 colours always enough?
The graph associated with a map is called a planar graph,because it can be drawn in the plane (2-dimensionalspace) without any edges crossing each other.
Question (1852):Is every planar graph 4-colourable?
Answer (Appel and Haken, 1976):Yes!
Answer (Robertson Sanders Seymour Thomas, ’95):Yes!
Answer (Werner and Gonthier, 2005):Yes!Why did they prove the Four Colour Theorem so manytimes?
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Are 4 colours always enough?
The graph associated with a map is called a planar graph,because it can be drawn in the plane (2-dimensionalspace) without any edges crossing each other.
Question (1852):Is every planar graph 4-colourable?
Answer (Appel and Haken, 1976):Yes!
Answer (Robertson Sanders Seymour Thomas, ’95):Yes!
Answer (Werner and Gonthier, 2005):Yes!Why did they prove the Four Colour Theorem so manytimes?
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Are 4 colours always enough?
The graph associated with a map is called a planar graph,because it can be drawn in the plane (2-dimensionalspace) without any edges crossing each other.
Question (1852):Is every planar graph 4-colourable?
Answer (Appel and Haken, 1976):Yes!
Answer (Robertson Sanders Seymour Thomas, ’95):Yes!
Answer (Werner and Gonthier, 2005):Yes!
Why did they prove the Four Colour Theorem so manytimes?
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Are 4 colours always enough?
The graph associated with a map is called a planar graph,because it can be drawn in the plane (2-dimensionalspace) without any edges crossing each other.
Question (1852):Is every planar graph 4-colourable?
Answer (Appel and Haken, 1976):Yes!
Answer (Robertson Sanders Seymour Thomas, ’95):Yes!
Answer (Werner and Gonthier, 2005):Yes!Why did they prove the Four Colour Theorem so manytimes?
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4 Colour Theorem: The proof
Four Colour Theorem: 1976, 1995, 2005Every planar graph (map) can be coloured with 4 colours.
"A proof is a proof. What kind of a proof? It’s a proof. Aproof is a proof, and when you have a good proof, it’sbecause it’s proven." – Jean Chrétien
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4 Colour Theorem: The proof
É 1976 proof was by computer.They proved the theorem by looking at nearly 2000configurations.The computation took more than a month.
É 1995 proof was also by computer.They reduced the proof to about 600 configurations.
É 2005 proof was generated by a computer system thatfinds mathematical proofs.The theorem is too complicated to prove by hand!
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4 Colour Theorem: The proof
Appel and Haken at work
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?
É This is the chromatic number of G, written χ(G)(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.
(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
We wish to colour a graph G, whose vertices are in the setV and whose edges are in the set E.
É A colouring of G is proper if no two adjacent verticesget the same colour.
É What is the minimum number of colours we need?É This is the chromatic number of G, written χ(G)
(χ is chi, the Greek letter).
Computing χ is an NP complete problem.(This means it takes a long time — we think!)
Question: Is P = NP?Translation: Can we solve NP-complete problems quicklywith a “normal” computer?
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Graph colouring
What can we say about χ(G), the chromatic number of agraph?
Is there any structure in G that forces us to use manycolours? (Lower bound on χ)
Can we give an upper bound on χ?
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Graph colouring
What can we say about χ(G), the chromatic number of agraph?
Is there any structure in G that forces us to use manycolours? (Lower bound on χ)
Can we give an upper bound on χ?
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Graph colouring
What can we say about χ(G), the chromatic number of agraph?
Is there any structure in G that forces us to use manycolours? (Lower bound on χ)
Can we give an upper bound on χ?
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Graph colouringSome more vocabulary:
É If v is a vertex, then any vertex adjacent to v is aneighbour of v.
É The degree of v is the number of neighbours of v.
É The maximum degree of a graph G is the highestdegree of a vertex in G.
É A clique in G is a set of vertices that are all connectedto each other.
A clique of size 4 in a map
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Graph colouringSome more vocabulary:
É If v is a vertex, then any vertex adjacent to v is aneighbour of v.
É The degree of v is the number of neighbours of v.
É The maximum degree of a graph G is the highestdegree of a vertex in G.
É A clique in G is a set of vertices that are all connectedto each other.
A clique of size 4 in a map
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Graph colouringSome more vocabulary:
É If v is a vertex, then any vertex adjacent to v is aneighbour of v.
É The degree of v is the number of neighbours of v.
É The maximum degree of a graph G is the highestdegree of a vertex in G.
É A clique in G is a set of vertices that are all connectedto each other.
A clique of size 4 in a map
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Graph colouringSome more vocabulary:
É If v is a vertex, then any vertex adjacent to v is aneighbour of v.
É The degree of v is the number of neighbours of v.
É The maximum degree of a graph G is the highestdegree of a vertex in G.
É A clique in G is a set of vertices that are all connectedto each other.
A clique of size 4 in a map
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Graph colouring
What is the largest clique you can find?
What is the highest degree you can find?What is the lowest degree you can find?
3 ≤ χ(G) ≤ 9
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Graph colouring
What is the largest clique you can find?What is the highest degree you can find?
What is the lowest degree you can find?3 ≤ χ(G) ≤ 9
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Graph colouring
What is the largest clique you can find?What is the highest degree you can find?What is the lowest degree you can find?
3 ≤ χ(G) ≤ 9
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Graph colouring
What is the largest clique you can find?What is the highest degree you can find?What is the lowest degree you can find?
3 ≤ χ(G) ≤ 9
![Page 93: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/93.jpg)
Graph colouring
Reed’s ConjectureThe chromatic number of any graph is at most theaverage of the clique number and the maximum degree,plus 1.
I wrote my Ph.D. dissertation on this problem.
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Sudoku: Graph colouring in disguise
1 4
1 3
3
How can we model sudoku as a graph colouring problem?
É Every square is a vertexÉ Some vertices are already coloured... we just need to
finish the colouring!É This problem is called precolouring extension.
![Page 95: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/95.jpg)
Sudoku: Graph colouring in disguise
1 4
1 3
3
How can we model sudoku as a graph colouring problem?
É Every square is a vertex
É Some vertices are already coloured... we just need tofinish the colouring!
É This problem is called precolouring extension.
![Page 96: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/96.jpg)
Sudoku: Graph colouring in disguise
1 4
1 3
3
How can we model sudoku as a graph colouring problem?
É Every square is a vertexÉ Some vertices are already coloured... we just need to
finish the colouring!
É This problem is called precolouring extension.
![Page 97: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/97.jpg)
Sudoku: Graph colouring in disguise
1 4
1 3
3
How can we model sudoku as a graph colouring problem?
É Every square is a vertexÉ Some vertices are already coloured... we just need to
finish the colouring!É This problem is called precolouring extension.
![Page 98: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/98.jpg)
Sudoku: Graph colouring in disguise
1 4
1 3
3
How can we model sudoku as a graph colouring problem?
É Every square is a vertexÉ Some vertices are already coloured... we just need to
finish the colouring!É This problem is called precolouring extension.
![Page 99: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/99.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 100: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/100.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 101: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/101.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 102: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/102.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 103: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/103.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 104: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/104.jpg)
The game chromatic number
Feeling competitive? Paint your opponent into a corner!
É There are k colours to choose from.
É Two players take turns colouring one vertex at a time.
É The colouring must always be proper.
É Player 1 wins if the graph can be coloured completely.
É Player 2 wins if there is a stalemate.
É Game chromatic number of G: Smallest k so thatPlayer 1 can always win.
![Page 105: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/105.jpg)
Colouring the plane
How many vertices can a graph have? Maybe ∞?
É Let every point in the plane be a vertex.
É Two points are connected if the distance betweenthem is exactly 1.
É How many colours do we need to colour the plane?É If two points are 1 apart, then they get different
colours.É Can you say why you need at least 3 colours?
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Colouring the plane
How many vertices can a graph have? Maybe ∞?
É Let every point in the plane be a vertex.É Two points are connected if the distance between
them is exactly 1.
É How many colours do we need to colour the plane?É If two points are 1 apart, then they get different
colours.É Can you say why you need at least 3 colours?
![Page 107: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/107.jpg)
Colouring the plane
How many vertices can a graph have? Maybe ∞?
É Let every point in the plane be a vertex.É Two points are connected if the distance between
them is exactly 1.É How many colours do we need to colour the plane?
É If two points are 1 apart, then they get differentcolours.
É Can you say why you need at least 3 colours?
![Page 108: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/108.jpg)
Colouring the plane
How many vertices can a graph have? Maybe ∞?
É Let every point in the plane be a vertex.É Two points are connected if the distance between
them is exactly 1.É How many colours do we need to colour the plane?É If two points are 1 apart, then they get different
colours.
É Can you say why you need at least 3 colours?
![Page 109: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/109.jpg)
Colouring the plane
How many vertices can a graph have? Maybe ∞?
É Let every point in the plane be a vertex.É Two points are connected if the distance between
them is exactly 1.É How many colours do we need to colour the plane?É If two points are 1 apart, then they get different
colours.É Can you say why you need at least 3 colours?
![Page 110: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/110.jpg)
Colouring the plane
Actually you need at least 4.This graph is called the Moser spindle.
But the bottom edge is much longer!
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Colouring the plane
You need between 4 and 7 colours.
É Nobody knows what the answer is.É If we change the rules a little bit, we need either 6 or
7 colours. But still, nobody knows.
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Colouring the plane
You need between 4 and 7 colours.
É Nobody knows what the answer is.
É If we change the rules a little bit, we need either 6 or7 colours. But still, nobody knows.
![Page 113: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/113.jpg)
Colouring the plane
You need between 4 and 7 colours.
É Nobody knows what the answer is.É If we change the rules a little bit, we need either 6 or
7 colours. But still, nobody knows.
![Page 114: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/114.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.
He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 115: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/115.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 116: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/116.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 117: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/117.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.
É Can your team figure it out?É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 118: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/118.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?
É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 119: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/119.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?É It may be difficult to explain.
É Start with piles of size 1, then 1 or 2, and try to findthe pattern.
![Page 120: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/120.jpg)
Bonus round: The game of NIM
He who laughs last laughs loudest.He who plays last loses. (This is a misère game)
É There are three piles of stones.É Two players take turns, choosing a pile and removingat least one stone.
É Whoever removes the last stone loses.
There is a very clever strategy.É Can your team figure it out?É It may be difficult to explain.É Start with piles of size 1, then 1 or 2, and try to find
the pattern.
![Page 121: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/121.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 122: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/122.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000
É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 123: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/123.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001
É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 124: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/124.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010
É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 125: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/125.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011
É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 126: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/126.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100
É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 127: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/127.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101
É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 128: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/128.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110
É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 129: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/129.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 130: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/130.jpg)
Bonus round: The game of NIM
Binary numbers: Let’s write the numbers 0 to 7 inbinary.
É 0 = 000É 1 = 20 = 001É 2 = 21 = 010É 3 = 21 + 20 = 011É 4 = 22 = 100É 5 = 22 + 20 = 101É 6 = 22 + 21 = 110É 7 = 22 + 21 + 20 = 111
We want to add them without carrying.In this world, the only numbers are 0 and 1, and1+1 = 0.
![Page 131: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/131.jpg)
Bonus round: The game of NIM
Binary numbers: Normal sum and Nim sum.
É In binary, the sum of 010 and 011 is 101.É (in decimal, we are saying 2 + 3 = 5).É But the Nim sum of 010 and 011 is 001,
because we don’t carry the 1.
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
![Page 132: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/132.jpg)
Bonus round: The game of NIM
Binary numbers: Normal sum and Nim sum.
É In binary, the sum of 010 and 011 is 101.
É (in decimal, we are saying 2 + 3 = 5).É But the Nim sum of 010 and 011 is 001,
because we don’t carry the 1.
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
![Page 133: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/133.jpg)
Bonus round: The game of NIM
Binary numbers: Normal sum and Nim sum.
É In binary, the sum of 010 and 011 is 101.É (in decimal, we are saying 2 + 3 = 5).
É But the Nim sum of 010 and 011 is 001,because we don’t carry the 1.
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
![Page 134: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/134.jpg)
Bonus round: The game of NIM
Binary numbers: Normal sum and Nim sum.
É In binary, the sum of 010 and 011 is 101.É (in decimal, we are saying 2 + 3 = 5).É But the Nim sum of 010 and 011 is 001,
because we don’t carry the 1.
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
![Page 135: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/135.jpg)
Bonus round: The game of NIM
Binary numbers: Normal sum and Nim sum.
É In binary, the sum of 010 and 011 is 101.É (in decimal, we are saying 2 + 3 = 5).É But the Nim sum of 010 and 011 is 001,
because we don’t carry the 1.
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
![Page 136: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/136.jpg)
Bonus round: The game of NIM
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
É If the Nim sum is not zero at the beginning of yourturn, can you always make it zero at the end of yourturn?
É If the Nim sum is zero at the beginning of your turn,can you make it nonzero at the end of your turn?
So Player 1 can win precisely if the Nim sum is not zero atthe beginning of the game.
![Page 137: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/137.jpg)
Bonus round: The game of NIM
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
É If the Nim sum is not zero at the beginning of yourturn, can you always make it zero at the end of yourturn?
É If the Nim sum is zero at the beginning of your turn,can you make it nonzero at the end of your turn?
So Player 1 can win precisely if the Nim sum is not zero atthe beginning of the game.
![Page 138: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/138.jpg)
Bonus round: The game of NIM
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
É If the Nim sum is not zero at the beginning of yourturn, can you always make it zero at the end of yourturn?
É If the Nim sum is zero at the beginning of your turn,can you make it nonzero at the end of your turn?
So Player 1 can win precisely if the Nim sum is not zero atthe beginning of the game.
![Page 139: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/139.jpg)
Bonus round: The game of NIM
The strategy:
É When you finish your turn, you want the Nim sum ofthe sizes of the piles to be zero.
É If you can do this, your opponent can’t do the samething to you.
É You want to do this until you can leave your opponentwith an odd number of piles with only one stone.
É If the Nim sum is not zero at the beginning of yourturn, can you always make it zero at the end of yourturn?
É If the Nim sum is zero at the beginning of your turn,can you make it nonzero at the end of your turn?
So Player 1 can win precisely if the Nim sum is not zero atthe beginning of the game.
![Page 140: Andrew D. Kingandrewdouglasking.com.s3-website-us-west-2.amazonaws.com/... · Andrew D. King Simon Fraser University, Burnaby, B.C. A taste of π, December 1, 2012. A little about](https://reader034.fdocument.pub/reader034/viewer/2022051804/5fecb91d5cd383416b5b881e/html5/thumbnails/140.jpg)
More 4-colouring maps
4-colouring a planar graph
http://www.nikoli.com/en/take_a_break/four_color_problem/