Math576: Combinatorial Game Theory Lecture note I
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1 Math576: Combinatorial Game Theory Lecture note I Linyuan Lu University of South Carolina Fall, 2018
2 Artificial Intelligence In 1996 Deep Blue (IBM) played against Garry Kasparov: won one, drew two, and lost three. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 2 / 47
3 Artificial Intelligence In 1996 Deep Blue (IBM) played against Garry Kasparov: won one, drew two, and lost three. In 1997, after heavily upgraded, Deep Blue won six-game rematch 3.5 : 2.5 against Kasparov. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 2 / 47
4 Artificial Intelligence In 1996 Deep Blue (IBM) played against Garry Kasparov: won one, drew two, and lost three. In 1997, after heavily upgraded, Deep Blue won six-game rematch 3.5 : 2.5 against Kasparov. in 2016, AlphaGo (Google) beat Lee Sedol in a five-game match. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 2 / 47
5 Artificial Intelligence In 1996 Deep Blue (IBM) played against Garry Kasparov: won one, drew two, and lost three. In 1997, after heavily upgraded, Deep Blue won six-game rematch 3.5 : 2.5 against Kasparov. in 2016, AlphaGo (Google) beat Lee Sedol in a five-game match. Will AI outsmart human being? How soon? How to play games smarter? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 2 / 47
6 About the course Textbook: Course Material Chapter 1-5, part of Chapter 7. Conways Game of Life Puzzles Assessment Homework Two midterm exam Final project Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 3 / 47
7 Disclaimer The slides are solely for the convenience of the students who are taking this course. The students should buy the textbook. The copyright of many figures in the slides belong to the authors of the textbook: Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 4 / 47
8 What Games? Number of players? Type of Games? Rules? Ending positions? Winning Strategies? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 5 / 47
9 Blue-Red Hackenbush Two players: Left and Right. Game board: blue-red graphs connected to the ground. Rules: Two players take turns. Right deletes one red edge and also remove any piece no longer connected to the ground. Left does the similar move but deletes one blue edge. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 6 / 47
10 Copy strategy Who wins? Tweedledum Tweedeldee (I) Tweedledum Tweedeldee (II) Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 7 / 47
11 Copy strategy Who wins? Tweedledum Tweedeldee (I) Tweedledum Tweedeldee (II) The second player has a winning strategy: copy the move of the first player. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 7 / 47
12 Zero Game A game have the value 0 if the second player has a winning strategy. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 8 / 47
13 Zero Game A game have the value 0 if the second player has a winning strategy. The sum of two games G and H, denoted by G+H, is a game that player can choose one of the game board to play at his/her turn. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 8 / 47
14 Zero Game A game have the value 0 if the second player has a winning strategy. The sum of two games G and H, denoted by G+H, is a game that player can choose one of the game board to play at his/her turn. For any game G, let G be the mirror image of G. Then G+( G) = 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 8 / 47
15 Zero Game A game have the value 0 if the second player has a winning strategy. The sum of two games G and H, denoted by G+H, is a game that player can choose one of the game board to play at his/her turn. For any game G, let G be the mirror image of G. Then G+( G) = 0. Two games G and H have the same value if G+( H) = 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 8 / 47
16 Game Values Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 9 / 47
17 Game Values What are the values of the following games? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 9 / 47
18 Answers Observation: If each edge in a red-blue Hackenbush game G is connected to the ground via its own color, then the other player cannot delete its opponent s edges. Therefore the value of G is the number of blue edges minus the number of red edges. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 10 / 47
19 Answers Observation: If each edge in a red-blue Hackenbush game G is connected to the ground via its own color, then the other player cannot delete its opponent s edges. Therefore the value of G is the number of blue edges minus the number of red edges = = 2 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 10 / 47
20 Half move 1 2 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 11 / 47
21 Half move Show that the following game is a zero-game. 1 2 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 11 / 47
22 Brace notation ւ Left first ց Right first 0 1 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 12 / 47
23 Brace notation ւ Left first ց Right first 0 1 {0 1} = 1 2 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 12 / 47
24 Brace notation More notation: ւ ց Left first Right first 0 1 {0 1} = 1 2 { } = 0 {0 } = 1 {1 } = 2 { 0} = 1 { 1} = 2 {n } = n+1 {n n+1} = n+ 1 2 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 12 / 47
25 More game values What are the values of the following games? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 13 / 47
26 More game values What are the values of the following games? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 13 / 47
27 A game of Ski-Jumps Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 14 / 47
28 A game of Ski-Jumps Two players: Left and Right. Game board: several skiers on a rectangular board Rules: Two players take turns. Left may move any skier a square or more Eastwards, or Right any one of his, Westwards, provided there is no active skier in the way. Such a move may take a skier off the slope; in this case he takes no further part in the game. Alternatively a skier on the square immediately above one containing a skier of the opposing team, may jump over him on the the square immediately below, provided this is empty. A man jumped over will never jump over anyone else. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 15 / 47
29 Some examples of Ski-Jumps Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 16 / 47
30 Some examples of Ski-Jumps 5 3 = 2 {2 3} = Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 16 / 47
31 Some examples of Ski-Jumps 5 3 = 2 {2 3} = Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 16 / 47
32 Some examples of Ski-Jumps 5 3 = 2 {2 3} = =-1 Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 16 / 47
33 A 3 5 board Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 17 / 47
34 Don t take the average! { } = 3 Why? Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 18 / 47
35 Simplicity Rule If the options in {a,b,c,... d,e,f,...} are all numbers, we say the number x fits if x is greater than each of a,b,c,... and less than each of d,e,f,... Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 19 / 47
36 Simplicity Rule If the options in {a,b,c,... d,e,f,...} are all numbers, we say the number x fits if x is greater than each of a,b,c,... and less than each of d,e,f,... Simplicity Rule: If there s any number that fits, the answer s the simplest number that fits. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 19 / 47
37 Simplicity Rule If the options in {a,b,c,... d,e,f,...} are all numbers, we say the number x fits if x is greater than each of a,b,c,... and less than each of d,e,f,... Simplicity Rule: If there s any number that fits, the answer s the simplest number that fits. For example, {0 1} = 1 2, { 1 2 1} = 3 4, { } = 0, { } = 3. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 19 / 47
38 Simplest Forms for Numbers Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 20 / 47
39 Toads-and-Frogs Two players: Left and Right. Game board: Some Toads and Frogs on a rectangular board. Rules: Two players take turns. Left moves one of Toads Eastwards. Right moves one of Frogs Westwards. The creature (Toad or Frog) may jump over an opposing creature, onto an empty square. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 21 / 47
40 An example Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 22 / 47
41 Game Values Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 23 / 47
42 Working out a horse Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 24 / 47
43 Game of CutCake Two players: Left and Right. Game board: A rectangular cake. Rules: Two players take turns. Left may cut any rectangle into two smaller ones along the North-South lines while Right cut it along the the East-West lines. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 25 / 47
44 Game Values in Cutcake Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 26 / 47
45 Maundy Cake Two players: Left and Right. Game board: A rectangular cake. Rules: Two players take turns. Left may cut any rectangle into any number of smaller equal ones along the North-South lines while Right cut it along the the East-West lines. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 27 / 47
46 Maundy Cake Two players: Left and Right. Game board: A rectangular cake. Rules: Two players take turns. Left may cut any rectangle into any number of smaller equal ones along the North-South lines while Right cut it along the the East-West lines. Ending positions: Whoever gets stuck is the loser. For example, a 4 9 cake may be cut into nine 4 1 or three 4 3 by Left; four 1 9 or two 2 9 by Right. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 27 / 47
47 Maundy Cake Values Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 28 / 47
48 Working out Maundy Cake Let M(r,l) be the value of Maundy Cake of dimension r l. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 29 / 47
49 Working out Maundy Cake Let M(r,l) be the value of Maundy Cake of dimension r l. r = 999: 333: 111: 37: 1 l = 1000: 500: 250: 125: 25: 5: 1 M(999,1000) = 5+1. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 29 / 47
50 Working out Maundy Cake Let M(r,l) be the value of Maundy Cake of dimension r l. r = 999: 333: 111: 37: 1 l = 1000: 500: 250: 125: 25: 5: 1 M(999,1000) = 5+1. r = 1000: 500: 250: 125: 25: 5: 1 l = 1001: 143: 13: 1 M(1000,1001) = ( 25)+( 5)+( 1) = 31. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 29 / 47
51 Four possible game outputs G > 0 or G is positive if player Left can always win. G < 0 or G is negative if player Right can always win. G = 0 or G is zero if second player can always win. G 0 or G is fuzz if first player can always win. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 30 / 47
52 Combination of outputs G 0 means that Left has a winning strategy provided Right plays first. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 31 / 47
53 Combination of outputs G 0 means that Left has a winning strategy provided Right plays first. G 0 means that Right has a winning strategy provided Left plays first. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 31 / 47
54 Combination of outputs G 0 means that Left has a winning strategy provided Right plays first. G 0 means that Right has a winning strategy provided Left plays first. G 0 means that Left has a winning strategy provided Left plays first. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 31 / 47
55 Combination of outputs G 0 means that Left has a winning strategy provided Right plays first. G 0 means that Right has a winning strategy provided Left plays first. G 0 means that Left has a winning strategy provided Left plays first. G 0 means that Right has a winning strategy provided Right plays first. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 31 / 47
56 Hackenbush Hotchpotch Two players: Left and Right. Game board: blue-red-green graphs connected to the ground. Rules: Two players take turns. Right deletes one red edge or one green edge and also remove any piece no longer connected to the ground. Left does the similar move but deletes one blue edge or one green edge. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 32 / 47
57 Sum of arbitrary games Let G L be the typical left options and G R be the typical right options. Then G = {G L G R }. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 33 / 47
58 Sum of arbitrary games Let G L be the typical left options and G R be the typical right options. Then G = {G L G R }. For two arbitrary games G = {G L G R } and H = {H L H R }, the sum of the games is defined as G+H = {G L +H,G+H L G R +H,G+H R }. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 33 / 47
59 Some properties If G 0 and H 0 then G+H 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 34 / 47
60 Some properties If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 34 / 47
61 Some properties If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 34 / 47
62 Some properties If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 34 / 47
63 Some properties If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. If G 0 and H 0 then G+H 0. We only prove the first property here. Assume Right plays first. If Right plays on G, then Left responds in G since Left has a winning strategy in G. If Right plays on H, then Left responds in H since Left has a winning strategy in H. In ether case, Left can win. Thus, G+H 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 34 / 47
64 Outcome of sum of games H = 0 H > 0 H < 0 H 0 G = 0 G+H = 0 G+H > 0 G+H < 0 G+H 0 G > 0 G+H > 0 G+H > 0 G+H?0 G+H 0 G < 0 G+H < 0 G+H?0 G+H < 0 G+H 0 G 0 G+H 0 G+H 0 G+H 0 G+H?0 Here G+H?0 are unrestricted. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 35 / 47
65 Comparing two games Use the negative game H = { H R H L }. One can define G = H, G > H, G < H, and G H. For example, define G > H if G+( H) > 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 36 / 47
66 Comparing two games Use the negative game H = { H R H L }. One can define G = H, G > H, G < H, and G H. For example, define G > H if G+( H) > 0. We have H = K H > K H < K H K G = H G = K G > K G < K G K G > H G > K G > K G?K G K G < H G < K G?K G < K G K G H G K G K G K G?K Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 36 / 47
67 Small Hackenbush Positions Flower is dwarfed by Very Small Hollyhocks of Either Sign. 1 2 n < flower < 1 2 n for any n. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 37 / 47
68 Small positive Hackenbush This house has a positive value but is smaller than any positive number. 0 < house < 1 2 n for any n. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 38 / 47
69 The game value = {0 0}. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 39 / 47
70 The game value How big is the star? = {0 0}. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 39 / 47
71 The game value = {0 0}. How big is the star? is less than any positive number, greater than any negative number, and fuzz with 0. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 39 / 47
72 The game value = {0 0}. How big is the star? is less than any positive number, greater than any negative number, and fuzz with 0. For any number x, let x = x+. We have {x x} = x. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 39 / 47
73 Game of Col Two players: Left and Right. Game board: a partial colored planar map. Rules: Two players take turns to paint regions of the map. Each player, when in his turn to move, paint one region of the map, Left using the color blue and Right using the color red. No two regions having a common frontier may be painted in the same color. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 40 / 47
74 Example 1 of Col Game The region belonging to Left only is blue-tinted while the one belonging to Right only is red-tinted. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 41 / 47
75 Example 2 of Col Game Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 42 / 47
76 Example 2 of Col Game {, 1,1 1} = {1 1} = 1. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 42 / 47
77 Alternative version of Col We can also represent regions by nodes and adjacency by edges. : nodes available for both Left and Right. : nodes available for Left only. : nodes available for Right only. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 43 / 47
78 Some Col Values Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 44 / 47
79 A theorem of Col Theorem Every position of Col has the value z or z for some number z. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 45 / 47
80 A theorem of Col Theorem Every position of Col has the value z or z for some number z. Proof: It is sufficient to show G L + G G R +. The statement follows from induction. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 45 / 47
81 Seating Couples Two players: Left and Right. Game board: some dinning tables of various sizes. Rules: Two players take turns to seat couples for a dinner. Left prefers to seat a lady to the left of her partner, while Right thinks it proper only to seat her to the right. No gentleman may be seated next to a lady other than his own partner. Ending positions: Whoever gets stuck is the loser. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 46 / 47
82 Values of seating couples LnL, a row of n empty chairs between two of Left s guests, RnR, a row of n empty chairs between two of Right s, and LnR or RnL, a row of n empty chairs between one of Left s guests and one of Right s. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 47 / 47
83 Values of seating couples LnL, a row of n empty chairs between two of Left s guests, RnR, a row of n empty chairs between two of Right s, and LnR or RnL, a row of n empty chairs between one of Left s guests and one of Right s. Recursive formular: LnL = {LaL+LbL LaR+RbL} RnR = {RaL+LbR RaR+RbR} = LnL LnR = {LaL+LbR LaR+RbR} = RnL. Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 47 / 47
84 Values of seating couples LnL, a row of n empty chairs between two of Left s guests, RnR, a row of n empty chairs between two of Right s, and LnR or RnL, a row of n empty chairs between one of Left s guests and one of Right s. Recursive formular: LnL = {LaL+LbL LaR+RbL} RnR = {RaL+LbR RaR+RbR} = LnL LnR = {LaL+LbR LaR+RbR} = RnL. n LnL LnR RnR Math576: Combinatorial Game Theory Linyuan Lu, University of South Carolina 47 / 47
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