the game of contorted fractions. 2 rules of the game typical position has a number of real numbers...
Post on 20-Dec-2015
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The Game of Contorted Fractions
2
Rules of the Game
Typical position has a number of real numbers in boxes.
The typical legal move is to alter just one of these numbers.
The number replacing a given one must have a strictly smaller denominator, or
If the given number is already an integer, the new number must be an integer strictly smaller in absolute value.
3
Rules of the Game
The only legal move for player 1, Left, is to decrease the number
The only legal move for player 2, Right, is to increase the number
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25
Possible Left options: 1/3, ¼, 0, -2, etc.Possible Right options: ½, 2/3, ¾, 17 ¼, etc.
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25
Left’s best option: 1/3
Right’s best option: ½
6
Notice
The game will come to an end when each fraction reaches zero, since we require that:
The number replacing a given one must have a strictly smaller denominator, or
If the given number is already an integer, the new number must be an integer strictly smaller in absolute value.
7
25
-23
37
8
01
10
11
12
21
13
23
32
31
14
25
35
34
43
53
52
41
15
27
38
37
47
58
57
45
54
75
85
74
73
83
72
51
9
25
-23
37
Combinatorial Game Theory
It’s All AboutHackenbush
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Blue-Red Hackenbush is an example of a game with 2 players, Left and Right.
In Blue-Red Hackenbush, Left deletes any bLue edge, and Right deletes any Red edge.
Whichever player cannot move, loses.
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Games
27
Defined
We are interested in games that:
Have just two players, often called Left and Right.
Have several, usually finitely many, positions, and often a particular starting position.
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There are clearly defined rules that specify the moves that either player can make from a given position to its options.
Have Left and Right moving alternately, in the game as a whole.
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In the normal play convention, has the first player unable to move losing.
Will always come to an end because some player will be unable to move.
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Allow both players to know what is going on; i.e. there is perfect or complete information.
Have no chance moves such as rolling dice or shuffling cards.
“Whose Game?”
*Pun from Winning Ways
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Outcome Classes
If Left starts
If
Right
starts
Left
Wins
Right
Wins
Left
Wins
Positive
(L wins)
Zero
(2nd wins)
Right
Wins
Fuzzy
(1st wins)
Negative
(R wins)
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Fuzzy Games
A fuzzy game is • not > 0, • not < 0,• and not = 0
A fuzzy game is confused with 0
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Fuzzy Games
-2 -1 1 2
G
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Games and Numbers
All numbers are games
But not all games are numbers!!!
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Positive, Negative, and Zero
Theoretic game value is determined using
Surreal Numbers
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Surrealism in the Arts
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How to Count
The old way of counting:
1 2 3 4
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How to Count
The new way of counting:
0 1 2 3
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How to Count
So we have 0, 1, 2, 3 = 4 bananas!!
Surreal Numbers
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Surreal Numbers: A Class
The Surreal Numbers are a proper Class of numbers.
Other proper Classes we already know: - The Natural Numbers - The Integers - The Rational Numbers - The Real Numbers - The Complex Numbers
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Construction
{ L | R }
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Construction
{ | }
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Construction
{ | }
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Construction
{ | } = 0
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The Zero Game of Hackenbush
You go first, I insist!!!
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Construction
{ 0 | } = 1
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Construction
{ | 0 } = -1
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{0 | } = 1 { | 0} = -1
{-1 | 1} = 0
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Red-Blue Hackenbush Chains
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1
2
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a
2
b
0
c
- ½
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d
e
f
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g ih
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j k
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l m
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n o
So, what’s the value of that loooooong chain???
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Berlekamp’s Rule
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Berlekamp’s Rule
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Berlekamp’s Rule
-4
.0 1 1 0 01 1
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Berlekamp’s Rule
-4
.0 1 1 0 01 1
-4 . 0 1 1 0 1 0 1
-(4 + 14
+ 1 8
+ 132
+ 1 ) 128
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Berlekamp’s Rule
-4
.0 1 1 0 01 1
-4 . 0 1 1 0 1 0 1
-(4 + 14
+ 1 8
+ 132
+ 1 ) 128
= - 4 53 128
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Berlekamp’s Rule
Likewise, we can convert a given fractional number into a Hackenbush string
Write the fractional number in binary 3 5/8 = 3.101
Replace the integer part with LLL (since positive value)
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Berlekamp’s Rule
Replace the point with LR (since positive)
Convert 1’s and 0’s thereafter into L’s and R’s,
but omitting the final 1 1 0 1 L R
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Berlekamp’s Rule
Result: 3 5/8 = 3 . 1 0 1 L L L L R L R
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Berlekamp’s Rule
This also works for real numbers that don’t terminate, except that there is no final 1 to be omitted 1/3 = 0 . 0 1 0 1 0 1 0 1 0 . . . LR R L R L R L R L R . . .
Value of a Game ofContorted Fractions
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Use Continued Fractions
We can use continued fractions, together with Berlekamp’s rule, to determine the game value of any fraction
We convert the fraction into a continued fraction, and then read off the partial quotients a1, a2, …, as alternate numbers of 0’s and 1’s, except that the first 0 is replaced by a binary point.
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Example
83 13 1 1 1 12 2
35 35 2 1 2 4
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Example
83 13 1 1 1 12 2
35 35 2 1 2 4
83
3579
2.01001111 2256
75
25
-23
37
76
25
-23
37
3 1 10
7 2 3
2 1 1
05 2 2
2 1 10
3 1 2
77
25
-23
37
3 1 10
7 2 3
37
70.0111
16
2 1 10
5 2 2
25
30.011
8
2 1 10
3 1 2
-23
30.11
4
78
25
-23
37
7 3 3 1
16 8 4 16
Positive Game Value: Left Wins
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Ready to Play??
Here’s a link to a nice little applet that will let you play against the computer
CONTORTED FRACTIONS