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The Wonderful World of Hackenbush Games And Their Relation to the Surreal Numbers

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The Men Behind the Magic: John H. Conway created the surreal numbers in 1969. Donald Knuth thought these numbers were dreamy and gave them their name: surreal numbers. “The surreal numbers include all the natural counting numbers, together with negative numbers, fractions, and irrational numbers, and numbers bigger than infinity and smaller than the smallest fraction.” A good way to get acquainted with these surreal numbers is via the Game of Hackenbush. ¼, e, sqrt(2), 0, -2, infintity, 1/infinity,

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grEen Hackenbush Rules: –Branches or lines which touch the “ground” or baseline. –Two players: Left and Right take turns making moves. –Either player can hack away a grEen branch. –A move consists of hacking away one of the segments, and removing that segment and all segments above it that are not connected to the ground. –Ground is considered as one node –Last person to hack wins. –Game Time: To the board…

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Hackenbush and Nim Three stalks = Nim piles of 3, 4, 5 Nim-sum of these is 3 + 4 + 5 = 2 Derive SG-value of 0 Is it a N or a P position?

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Properties of Hackenbush Trees Value of a continuous color is 1/2 n where n is the number of branches. Colon Principle: When branches come tgogether at a vertex, one may replace the branches by a non-branching stalk of length equal to their nim- sum. Fusion Principle: The vertices on any circuit may be fused without changing the Sprague- Grundy value of the graph. –Loops reduce to lines –Example: Girl to green shrub (via fusion) to blade of grass (via Colon) A.k.a. Great topics for the final question!!!

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Blue Red Hackenbush Same as Green Hackenbush except… –A partizan game –Red branches may only be hacked by Right. bLue branches only hackable by Left. Play game on board. –Tweedledee and Tweedledum I (modify one to have a lollypop (for fusion))

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Finding Values in Blue Red Hackenbush The value of the game is in terms of the number of moves in Right’s advantage. A negative value corresponds to a “negative advantage” to Right. A.k.a. an advantage to Left What does half a move advantage for Right look like?

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Notation for Surreal Numbers A generic representation –{X L |X R } = V X L is the amount of moves which Left has when he moves first. X R is the amount of moves which Right has when he moves first. Start counting moves at 0 Some examples: –{ | } = 0 –{0| }= 1 –{ |0}= -1 –{0|1} = {-1,0 | 1} = ½ –{1| } = {0,1| } = 2 All of these values represent the value for the Left player

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Using Hackenbush to Explore Surreal Numbers Further –Think of Hackenbush as another notation… Take a look at 2/3: –Think of this picture as a “visual limit”. –Imagine the picture that forms as a result of following the visual pattern for larger and larger hackenbush strings -The picture in your mind’s eye is very close to 2/3. - To calculate the value of the next hackenbush string. Take current hackenbush string length, n, calculate a value, 1/2n. Whether the next color in the pattern is red or blue respectively subtract or add that value to the value of the current string. 0 1 ½ ¾ 5/8 11/16 21/32 43/64 84/128 171/256 341/512 683/1024 1365/2048

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Using Hackenbush to Explore Surreal Numbers Further Part II Take a look at : –This is a hackenbush string which is infinite in length. –Convert to a binary number –Since its , there is no repeating pattern. 3.001001000011111101101010100010010000101101000 1…

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The Infinite Ordinal Numbers Omega is a really big number, similar to infinity. Omega is a hackenbush tree, all the same color with an infinite number of branches.

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Conclusions The Surreal Numbers encompass a very large scale. Hackenbush provides a game we can play with the surreal numbers More importantly hackenbush provides a way to visualize the surreal numbers. –Two players/sets Left and Right –A way to “see” numbers of infinite size

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