Hackenbush and the surreal numbers James A. Swenson University of Wisconsin{Platteville [email protected] September 28, 2017 Bi-State Math Colloquium James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 1 / 37 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 Epigraph James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 3 / 37 Epigraph Propositiones aliquot, que in Scholis Societatis non sunt docendæ... 25 Continuum successiuum & intensio qualitatum solis indiuisibilibus constant.. 30 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 1651. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 4 / 37 Epigraph Some propositions which must not be taught in the Society's schools... 25 The line of succession and of the intensity of qualities are made up of indivisible points.. 30 Infinity in multitude and infinity in magnitude can be enclosed between two units or two points. Ordinance for higher study. Sent by Our Most Rev- erend Holy Father Francisco Piccolomineo [Superior General of the Jesuit Order], year 1651. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 5 / 37 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 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 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-`a-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 http://www-groups.dcs.st-and.ac.uk/∼history/Biographies/Conway.html James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 8 / 37 Donald K. Knuth (1938{ ) James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 9 / 37 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 To move, delete an edge of your color, plus any edges no longer connected to the ground. Blue moves first. If it's your Red loses! turn and you can't move, you lose. 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 Blue moves first. If it's your Red loses! turn and you can't move, you lose. 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 Red loses! 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 Red loses! 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 Red loses! 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 Red loses! 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 Red loses! 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 Red loses! 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 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 Red loses! turn and you can't move, you lose. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 11 / 37 Game notation To play well, you need to know your options! 8 9 <> => = ; ; > > : ; James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 12 / 37 Let's improve our lives by giving this game a name: • = fjg. The simplest game 8 9 <> => = > > : ; James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 13 / 37 The simplest game 8 9 <> => = > > : ; Let's improve our lives by giving this game a name: • = fjg. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 13 / 37 In symbols, this game is f•|}. Let's name it: • = f•|}. 8 9 <> => = > > : ; In symbols, this game is fj •}. Let's name it: • = fj •}. The next simplest games 8 9 <> => = > > : ; James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37 8 9 <> => = > > : ; In symbols, this game is fj •}. Let's name it: • = fj •}. The next simplest games 8 9 <> => = > > : ; In symbols, this game is f•|}. Let's name it: • = f•|}. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37 The next simplest games 8 9 <> => = > > : ; In symbols, this game is f•|}. Let's name it: • = f•|}. 8 9 <> => = > > : ; In symbols, this game is fj •}. Let's name it: • = fj •}. James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 14 / 37 Games with up to two edges ^ • = fjg • = f•|} • = f•|} ! = f•; •|} • = f•| •} • = f•j •} • = fj •} • = fj •g < = fj •; •g • = f•j •} _ James A. Swenson (UWP) Hackenbush and the surreal numbers 9/14/17 15 / 37 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 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 Example Recall • = fjg. Since •L = ? = •R , it is vacuously true that • ≤ •. Order relation on games Definition Let G = fGLj GR g and H = fHLj HR g be games. This means that GL and GR 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 2 GL with H ≤ X ; and 2 there is no Y 2 HR 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 Order relation on games Definition Let G = fGLj GR g and H = fHLj HR g be games. This means that GL and GR 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 2 GL with H ≤ X ; and 2 there is no Y 2 HR with Y ≤ G.