It is not obvious, but no matter how randomized the board setup is, and how often the connections from the left to the right intersect (when you compare the connections between pairs of phones), each phone on the left is always paired to one single phone on the right and vice versa. Can you prove that this one-to-one relationship always holds?
When such connection problems are published in a newspaper, usually the connections consist of very curvy cables that have crossovers, but are always meant to be separate (in 3D) and do not use common parts of the connecting wires.
In such cases the one-to-one relationship is obvious and trivial.
This Demonstration shows a very different approach to pairwise connections, namely one where the connecting wires use an unlimited number of common parts. Here the one-to-one relationship is not obvious at all.
Setup pop-up menu
You have the choice of having 80 vertical bars or 120 vertical bars in your randomized setup.
To pick another phone on the left border, simply click one that is not red.
Resets the game to the starting position.
Randomizes the setup.
Fractal by the author; created with the Fractint freeware.
This game was originally published as the Zillions game "Phone 2" by the same author.
Sometimes this type of puzzle is presented tilted by 90 degrees, then a set of ladders is used that have to be climbed from the bottom to the top.