Hackenbush and the surreal numbers

Transcription

1 Hackenbush and the surreal numbers James A. Swenson University of Wisconsin Platteville September 28, 2017 Bi-State Math Colloquium James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 1 / 37

2 Thanks for coming! I hope you ll enjoy the talk; please feel free to get involved! James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 2 / 37

3 Epigraph James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 3 / 37

4 Epigraph Propositiones aliquot, que in Scholis Societatis non sunt docendæ Continuum successiuum & intensio qualitatum solis indiuisibilibus constant Infinitum in multitudine, & magnitudine potest claudi inter duas unitates, vel duo puncta. Ordinatio pro studiis superioribus.... A[dmodum] R[everendus] P[ater] N[oster] Francisco Piccolomineo ad Prouincias Missa Anno James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 4 / 37

5 Epigraph Some propositions which must not be taught in the Society s schools The line of succession and of the intensity of qualities are made up of indivisible points Infinity in multitude and infinity in magnitude can be enclosed between two units or two points. Ordinance for higher study. Sent by Our Most Reverend Holy Father Francisco Piccolomineo [Superior General of the Jesuit Order], year James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 5 / 37

6 Outline 1 Heroes 2 Games 3 Ordering of games 4 Surreal numbers James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 6 / 37

7 John H. Conway (1937- ) Conway is a world-famous, award-winning mathematician, who has been a professor at Cambridge and (currently) Princeton. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 7 / 37

8 John H. Conway (1937 ) Conway is incredibly untidy. The tables in his room at the Department of Pure Mathematics and Mathematical Statistics in Cambridge are heaped high with papers, books, unanswered letters, notes, models, charts, tables, diagrams, dead cups of coffee and an amazing assortment of bric-à-brac, which has overflowed most of the floor and all of the chairs, so that it is hard to take more than a pace or two into the room and impossible to sit down. If you can reach the blackboard there is a wide range of coloured chalk, but no space to write. His room in College is in a similar state. In spite of his excellent memory he often fails to find the piece of paper with the important result that he discovered some days before, and which is recorded nowhere else. Richard Guy, quoted at history/biographies/conway.html James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 8 / 37

9 Donald K. Knuth (1938 ) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 9 / 37

10 Outline 1 Heroes 2 Games 3 Ordering of games 4 Surreal numbers James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 10 / 37

11 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

12 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

13 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

14 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

15 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

16 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

17 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

18 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

19 The rules of Hackenbush Hackenbush is a game played by two players, Blue and Red, on a rooted graph with colored edges. To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it s your turn and you can t move, you lose. Red loses! James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37

20 Game notation To play well, you need to know your options! =,, James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 12 / 37

21 The simplest game = James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 13 / 37

22 The simplest game = Let s improve our lives by giving this game a name: = { }. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 13 / 37

23 The next simplest games = James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37

24 The next simplest games = In symbols, this game is { }. Let s name it: = { }. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37

25 The next simplest games = In symbols, this game is { }. Let s name it: = { }. = In symbols, this game is { }. Let s name it: = { }. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37

26 Games with up to two edges = { } = { } = { }! = {, } = { } = { } = { } = { } = {, } = { } James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 15 / 37

27 Outline 1 Heroes 2 Games 3 Ordering of games 4 Surreal numbers James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 16 / 37

28 Comparing games Idea If G and H are games, we want: G H when H is at least as good as G for Blue. ( ) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 17 / 37

29 Order relation on games Definition Let G = {G L G R } and H = {H L H R } be games. This means that G L and G R are sets of games smaller than G, etc., so the following definition is recursive, not circular: We say G H provided that: 1 there is no X G L with H X ; and 2 there is no Y H R with Y G. ( Blue can t make G into something as good as H, and Red can t make H into something as bad as G. ) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 18 / 37

30 Order relation on games Definition Let G = {G L G R } and H = {H L H R } be games. This means that G L and G R are sets of games smaller than G, etc., so the following definition is recursive, not circular: We say G H provided that: 1 there is no X G L with H X ; and 2 there is no Y H R with Y G. ( Blue can t make G into something as good as H, and Red can t make H into something as bad as G. ) Example Recall = { }. Since L = = R, it is vacuously true that. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 18 / 37

31 Comparison: adding a single edge Theorem Adding a blue edge makes a game better for Blue; adding a red edge makes it worse. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 19 / 37

32 Comparison: adding a single edge Theorem Adding a blue edge makes a game better for Blue; adding a red edge makes it worse. Corollary James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 19 / 37

33 Comparison: adding a single edge Theorem Adding a blue edge makes a game better for Blue; adding a red edge makes it worse. Corollary and and James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 19 / 37

34 Comparison: adding a single edge Theorem Adding a blue edge makes a game better for Blue; adding a red edge makes it worse. Proposition Corollary. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 19 / 37

35 Good news: is reflexive Theorem If G is a game, then G G. Proof (Induction on number of edges). We know. Now let G be a game with at least one edge. Suppose for (transfinite) induction that H H whenever H has fewer edges than G. Hence G G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 20 / 37

36 Good news: is reflexive Theorem If G is a game, then G G. Proof (Induction on number of edges). We know. Now let G be a game with at least one edge. Suppose for (transfinite) induction that H H whenever H has fewer edges than G. Sftsoc: G G. we reach a contradiction. Hence G G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 20 / 37

37 Good news: is reflexive Theorem If G is a game, then G G. Proof (Induction on number of edges). We know. Now let G be a game with at least one edge. Suppose for (transfinite) induction that H H whenever H has fewer edges than G. Sftsoc: G G. Then either there is some X G L with G X or there is some Y G R with Y G. In either case, we reach a contradiction. Hence G G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 20 / 37

38 Good news: is reflexive Theorem If G is a game, then G G. Proof (Induction on number of edges). We know. Now let G be a game with at least one edge. Suppose for (transfinite) induction that H H whenever H has fewer edges than G. Sftsoc: G G. Then either there is some X G L with G X or there is some Y G R with Y G. In the first case, since G X, there is no Z G L with X Z... but X G L and X X (by induction). In either case, we reach a contradiction. Hence G G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 20 / 37

39 Good news: is reflexive Theorem If G is a game, then G G. Proof (Induction on number of edges). We know. Now let G be a game with at least one edge. Suppose for (transfinite) induction that H H whenever H has fewer edges than G. Sftsoc: G G. Then either there is some X G L with G X or there is some Y G R with Y G. In the first case, since G X, there is no Z G L with X Z... but X G L and X X (by induction). In the second case, since Y G, there is no Z G R with Z Y... but Y G R and Y Y (by induction). In either case, we reach a contradiction. Hence G G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 20 / 37

40 More good news: is transitive Fact If G H and H K, then G K. Proof (Induction on total number of edges in G H K). Base case: ( ) ( ). Let G H K. 1 Sftsoc: X G L and K X, so H K X. By induction, H X, so G H. 2 Sftsoc: Y K R and Y G, so Y G H. By induction, Y H, so H K. Hence G K. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 21 / 37

41 Bad news: is not antisymmetric Proposition = { } = { } James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 22 / 37

42 Bad news: is not antisymmetric Proposition = { } = { } Proof. 1 Let X L. Then X = = { }. Now X R and, so X. R =. So. 2 L =. Let Y R. Then Y = = { }. Now Y L and, so Y. So. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 22 / 37

43 Games with up to two edges (, ) (, ) (,! ) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 23 / 37

44 Outline 1 Heroes 2 Games 3 Ordering of games 4 Surreal numbers James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 24 / 37

45 Forcing antisymmetry Definition Let G and H be games. We say G H provided that G H and H G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 25 / 37

46 Forcing antisymmetry Definition Let G and H be games. We say G H provided that G H and H G. Definition is an equivalence relation; a -equivalence class is called a surreal number. We denote the equivalence class of a game G = {G L G R } by [G] = [G L G R ]. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 25 / 37

47 Forcing antisymmetry Definition Let G and H be games. We say G H provided that G H and H G. Definition is an equivalence relation; a -equivalence class is called a surreal number. We denote the equivalence class of a game G = {G L G R } by [G] = [G L G R ]. Definition [ ] The number zero is 0 = = [ ] = [ ]. [ ] The number one is 1 = = [ ] = [ ] = [0 ] = [[ ] ]. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 25 / 37

48 Ordering and strategy Fact [G] < 0 Red can always win the game G. [G] = 0 the second player can always win the game G. 0 < [G] Blue can always win the game G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 26 / 37

49 Adding games Definition If G and H are games, G + H is the game you get by putting G and H next to each other. + = 1 + [ ] = [ ] = [ ] = 0 James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 27 / 37

50 Adding games Definition If G and H are games, G + H is the game you get by putting G and H next to each other. + = 1 + [ ] = [ ] = [ ] = 0 Definition [ ] The number negative one is 1 = = [ ] = [ ] = [ [ ]]. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 27 / 37

51 The opposite of a game Definition G is the game you get by flipping the color of each edge in G. G H is shorthand for G + ( H). = James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 28 / 37

52 The opposite of a game Definition G is the game you get by flipping the color of each edge in G. G H is shorthand for G + ( H). = Proposition ( G) = G. If G H, then H G. G + ( G) 0. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 28 / 37

53 Ordering and strategy Fact If G H, then G + K H + K. If G 1 G 2 and H 1 H 2, then G 1 H 1 G 2 H 2. Definition We say [G] [H] provided that G H. Corollary [G] < [H] Red can always win the game G H. [G] = [H] the second player can always win the game G H. [H] < [G] Blue can always win the game G H. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 29 / 37

54 G is between G L and G R Theorem Let G be a game. 1 If X G L, then X G. 2 If Y G R, then G Y. ( It would always be better to pass. ) Proof. 1 Suppose X G L is the result when Blue deletes the blue edge e from G. Now consider the game G X, and suppose it s Red s move. If Red deletes an edge from G that has a mirror image in X, or an edge from X that has a mirror image in G, then Blue responds by deleting that mirror image. Otherwise, Red deletes an edge from G with no mirror image in X, and Blue responds by deleting e. On Red s first turn after the deletion of e, the position has the form H H, so Blue can win (by mirroring Red). This shows that Blue can win G X, so [G X ] 0. Thus X G. 2 (Similar.) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 30 / 37

55 All games are comparable Theorem If G H, then H G. Proof. Suppose G H. We consider two cases. 1 Suppose X G L and H X. We know X G. By transitivity, H G. 2 Suppose Y H R and Y G. We know H Y. By transitivity, H G. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 31 / 37

56 What is [ ]? Example The game shown below is + + : The second player can always win, so [ ] [ ] + + ( 1) = 0. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 32 / 37

57 What is [ ]? Example The game shown below is + + : The second player can always win, so [ ] [ ] + + ( 1) = 0. Definition The number one half is 1 2 = [ ]. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 32 / 37

58 Games with up to two edges James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 33 / 37

59 Games with more edges π ε ω ω ω + 1 2ω James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 34 / 37

60 More game arithmetic Definition Suppose G and H are games with X G G L, Y G G R, X H H L, Y H H R. Then: {X G H + GX H X G X H, Y G H + GY H Y G Y H } (GH) L. {X G H + GY H X G Y H, Y G H + GX H Y G X H } (GH) R. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 35 / 37

61 More game arithmetic Definition Suppose G and H are games with X G G L, Y G G R, X H H L, Y H H R. Then: {X G H + GX H X G X H, Y G H + GY H Y G Y H } (GH) L. {X G H + GY H X G Y H, Y G H + GX H Y G X H } (GH) R. Definition Let G and H be games with [G] > 0 and [H] > 0.1 If, for every X G G L, Y G G R, X H H L, and Y H H R, we have { } 1 1 Y G (1 + (Y G G)X H ), X G (1 + (X G G)Y H ) H L and { } 1 1 X G (1 + (X G G)X H ), Y G (1 + (Y G G)Y H ) H R, then [G][H] = 1. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 35 / 37

62 The surreal numbers are universal Theorem Every ordered field is isomorphic to a subfield of the surreal numbers. The proof requires the axiom of global choice and applies only to sets.... James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 36 / 37

63 References [1] Amir Alexander, Infinitesimal: How a dangerous mathematical theory shaped the modern world, Scientific American/Farrar, Strauss, and Giroux, New York, [2] John H. Conway, On numbers and games, 2nd ed., A K Peters, Ltd., Natick, MA, [3] John H. Conway and Richard K. Guy, The book of numbers, Copernicus, New York, [4] Tom Davis, Hackenbush (December 15, 2011), [5] Philip Ehrlich, All numbers great and small, Real numbers, generalizations of the reals, and theories of continua, Synthese Lib., vol. 242, Kluwer Acad. Publ., Dordrecht, 1994, pp [6] Gretchen Grimm, An introduction to surreal numbers (May 8, 2012), [7] D. E. Knuth, Surreal numbers: how two ex-students turned on to pure mathematics and found total happiness, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, [8] Jonas Sjöstrand, Combinatorial game theory (March 2015), [9] Wikipedia contributors, Surreal number (July 6, 2017), Image sources Book cover: Conway photo: H Conway 2005.jpg Conway sketch: Fraser, Simon J. Knuth photo: Appelbaum, Jacob. Ordinatio: TikZ Diagrams in Math Mode. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 37 / 37