Ghost Leg

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 vertical lines and horizontal lines, which we refer to as legs. Very often the number of vertical lines is the same as the number of people playing, and at the bottom of each line there is an item - a thing that will be paired with a player. The number of horizontal lines can be zero or more, and they are drawn between adjacent vertical lines. The general rule for playing this game is: first choose a line on the top, and follow this line downwards. When a horizontal line (leg) is encountered, follow it to get to another vertical line and continue downwards. Repeat this procedure until reaching the end of the vertical line. Then the player is given the thing written at the bottom of the line.

If the elements written above the Ghost Leg are treated as a sequence, and after the Ghost Leg is used, the same elements are written at the bottom, then the starting sequence has been transformed to another permutation. Hence, Ghost Leg can be treated as a kind of permuting operator.

Process

As an example, consider assigning roles in a play to actors.

# 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 leg to the board. Each leg must connect two adjacent vertical lines, and must not touch any other horizontal line.
# Once this is done, a path is traced from top of each vertical line to the bottom. As you follow the line down, if you come across a leg, you must follow it to the adjacent vertical line on the left or right, then resume tracing down. You continue until you reach the bottom of a vertical line, and the top item you started from is now paired with the bottom item you ended on.

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.

Mathematics

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 unpredictable the final outcome is.

One way of realizing how this works is to consider the analogy of coins in cups. You have "n" coins in "n" cups, representing the items at the bottom of the amidakuji. Then, each leg that is added represents swapping the position of two adjacent cups. Thus, it is obvious that in the end there will still be "n" cups, and each cup will have one coin, regardless of how many swaps you perform.

Properties

Permutation

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 applying it a finite number of times, the output sequence will become identical to the original input sequence.

i.e. Let M be a matrix representing a particular ghost legMⁿ=Ifor some finite n

Reversibility

For any Ghost Leg M,there must exist a Ghost Leg M^-1,such that M M^-1=I

Odd/Even property of permutation

As each leg represents one permutation, the number of legs indicates the odd/even permutation property of the Ghost Leg.

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 Ghost Leg.There are an infinite number of Ghost Legs representing the same permutation.

Prime

As there are an infinite number of Ghost Legs representing a particular permutation, it is obvious those Ghost Legs have a kind of equivalence. Among those equivalent Ghost Legs, the one(ones) which have smallest number of legs are called Prime.

Bubble sort and Highest Simplicity

A Ghost Leg can be constructed arbitrarily, but such a Ghost Leg is not necessarily Prime.It can be proven that only those Ghost Legs constructed by Bubble sort contains the least number of legs, and hence is Prime. This is equivalent to saying that Bubble sort performs the minimum number of adjacent exchanges to sort a sequence.

Maximum number of legs of Prime

For a permutation with n elements, the maximum number of neighbor exchanging = $frac\{n(n-1)\}\{2\}$

In the same way, the maximum number of legs in a Prime with n Tracks = $frac\{n(n-1)\}\{2\}$

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 identities are repeatedly applied in order to move and eliminate "useless" legs.
# =
# =

When the two identities cannot be applied any more, the ghost leg is proved to be exactly the same as the Ghost Leg constructed by Bubble sort, thus Bubblization can reduce Ghost Legs to Primes.

Randomness

Ghost Leg is not a good random permutation generator. When the number of legs is fixed and the legs are added randomly, the probability of getting different permutation is uneven. For example, if the number of legs is 3 and number of tracks is 4, the probability of getting is larger than that of .

Games

Konami produced an arcade game named Amidar which uses an Amidakuji board and rules as a setting for a Pac-Man/Qix like game.

New Super Mario Bros. and feature an Amidakuji-style minigame in which the player uses the stylus to trace lines that will lead the character down the right path.

External links

* http://www.geocities.com/Athens/Acropolis/7247/amidakuji.html
* [http://www2.edc.org/makingmath/studentWork/amidaKuji/AmidaKujiByDavidSenft.pdf Ladders: A Research Paper by David Senft] (PDF)