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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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
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.
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
13
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.
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.
28
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.
29
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.
30
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.
32
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)
42
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
67
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)
68
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
70
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 . . .
72
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.