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'''Ghost Leg''' (Chinese: [[畫鬼腳]]), known in Japan as ''Amidakuji'' (阿弥陀籤), is a method of lottery designed to create random pairings between two sets of any number of things, as long as the number of elements in each set is the same. This is often used to distribute things among people, where the number of things distributed is the same as the number of people. For instance, chores or prizes could be assigned fairly and randomly this way.
It consists of some
If the elements written above the Ghost Leg are treated as a [[sequence]], and after the Ghost Leg,
==Process==
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# To start with, the two sets are enumerated horizontally across a board. The actors' names would go on top, and the roles on the bottom. Then, vertical lines are drawn connecting each actor with the role directly below it.
# The names of the actors and/or roles are then concealed so that people do not know which actor is on which line, or which role is on which line.
# Next, each actor adds a
# Once this is done,
Another process involves creating the ladder beforehand, then concealing it. Then people take turns choosing a path to start from at the top. If no part of the amidakuji is concealed, then it is possible to fix the system so that you are guaranteed to get a certain pairing, thus defeating the idea of random chance.
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Part of the appeal for this game is that, unlike random chance games like [[rock, paper, scissors]], amidakuji will always create a 1:1 correspondence, and can handle arbitrary numbers of pairings (although pairing sets with only two items each would be fairly boring). It is guaranteed that two items at the top will never have the same corresponding item at the bottom, nor will any item on the bottom ever lack a corresponding item at the top.
It also works regardless of how many horizontal lines are added. Each person could add one, two, three, or any number of lines, and the 1:1 correspondence would remain. The more lines that are added, the more
One way of realizing how this works is to consider analogy of coins in cups. You have ''n'' coins in ''n'' cups, representing the items at the bottom of the amidakuji. Then, each
==Properties==
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Ghost Leg can transform the sequence of the input elements into a different order at the output. Thus, it is a permutation operator.
===Periodicity===
For any Ghost Leg, after
i.e. Let '''M''' be a matrix representing a particular ghost leg
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for some finite n
===Reversibility===
For any
there must exist a
such that '''M''' '''M'''<sup>-1</sup>='''I'''
===Odd/Even property of permutation===
As each
i.e. odd number of legs = odd permutation
===Infinite Ghost Legs with same permutation===
Every permutation must be possible to be expressed as a Ghost Leg, but a particular permutation does not correspond to a unique
There are infinite number of Ghost Legs representing the same permutation.
==Prime==
As there are infinite number of Ghost Legs representing
===Bubble sort and Highest Simplicity===
Ghost Leg can be
It
===Maximum number of legs of Prime===
For a permutation with n elements, the maximum number of
In the same way, the maximum number of legs in a Prime with n Tracks = <math>\frac{n(n-1)}{2}</math>
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===Bubblization===
For an arbitrary Ghost Leg, it is possible to transform it into Prime by a procedure called '''Bubblization'''.
When Bubblization operates, the following 2
#[[Image:GhostLeg4.GIF|80 px]] = [[Image:GhostLeg5.GIF|80 px]]
#[[Image:GL6.GIF|60 px]] = [[Image:GhostLeg7.GIF|60 px]]
When the two identities cannot be applied any more, the ghost leg is proved to be exactly the same as the
==Randomness==
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