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.
“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
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…
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?
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 nimsum.
• Fusion Principle: The vertices on any circuit
may be fused without changing the SpragueGrundy value of the graph.
– Loops reduce to lines
– Example: Girl to green shrub (via fusion) to blade of
grass (via Colon)
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))
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?
Notation for Surreal Numbers
• A generic representation
– {XL|XR} = 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
{0|1} = {-1,0 | 1} = ½
{1| } = {0,1| } = 2
• All of these values represent the value for the Left player
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
0 blue
1 ½ ¾ 5/8 11/16 21/32 43/64 84/128 171/256 341/512
683/1024 1365/2048
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
…
w: The Infinite Ordinal Numbers
• Omega is a really big number, similar to
infinity.

ww
• Omega is a hackenbush tree, all the same
color with an infinite number of branches.
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