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The Wonderful World of Hackenbush Games

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

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The Wonderful World of Hackenbush Games

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

  2. 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, p, e, sqrt(2), 0, -2, infintity, 1/infinity, w

  3. 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…

  4. 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?

  5. Properties of Hackenbush Trees A.k.a. Great topics for the final question!!! • Value of a continuous color is 1/2n 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)

  6. BlueRed 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))

  7. Finding Values in BlueRed 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?

  8. Notation for Surreal Numbers • A generic representation • {XL|XR} = V • XLis the amount of moves which Left has when he moves first. • XRis 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

  9. 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 5/811/1621/32 43/64 84/128 171/256 341/512 683/10241365/2048

  10. Using Hackenbush to Explore Surreal Numbers Further Part II • Take a look at p: • This is a hackenbush string which is infinite in length. • Convert p to a binary number • Since its p, there is no repeating pattern. • 3.0010010000111111011010101000100100001011010001…

  11. w: The Infinite Ordinal Numbers • Omega is a really big number, similar to infinity. ¥ + 1 = ¥ w + 1 = w + 1 • Omega is a hackenbush tree, all the same color with an infinite number of branches.

  12. 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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