The Wonderful World of Hackenbush Games

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

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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
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)

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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 Rights 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
• XLXR V
• XL is the amount of moves which Left has when he
  moves first.
• XR is the amount of moves which Right has when he
  moves first.
• Start counting moves at 0
• Some examples
• 0
• 0 1
• 0 -1
• 01 -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 minds 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 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

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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.

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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