Amidakuji
Amidakuji is a system of pairing a set of objects to another set with the same number of elements, usually used for lottery and prize games. As shown above, each designated player will choose his or her top spot (A B C D) and will first follow the vertical line. Should they encounter a horizontal "leg," they will have to follow that path horizontally before resuming the vertical trail, and so on. (Two adjacent legs can not be at the same level.)
For 4 players, if 7 legs are going to be put among these 4 lines, is it possible for each of the 4 players to reach the bottom directly below them?
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1 solution
For a pair of adjacent lines, we need an even number of legs between them in order to bring the top players to the destination directly below them. In other words, a pair of legs between the adjacent lines will be equivalent to none at all, as shown below:
Thus, if we want to have the same results for all 4 players, we need to have even number of legs between all 3 gaps. That means, the total number of legs will have to be even. Therefore, with 7 legs, we can not achieve such outcome.