Amidakuji
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Amidakuji (阿弥陀籤) is a Japanese 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.
[edit] 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 horizontal line to the board. Each line must connect two adjacent vertical lines, and must not directly touch any other horizontal line.
- Once this is done, the vertical lines are traced from top to bottom. As you follow the line down, if you come across a horizontal line, you 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.
[edit] 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 random the final outcome is.
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 horizontal bar 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.
