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Amida framework.

Amida frameworks are compound ladders like the one shown in the diagram with any number of uprights and any number of horizontal rungs, no two rungs having a common endpoint. The winning position is marked at the foot of one of the uprights. A path is marked going downwards from the top of each upright. When coming to a rung the path goes across and continues on down the adjacent upright. A winning position is given at the foot of one of the uprights and in the illustration player number 2 wins.

Draw your own frameworks and fill in the positions where the players finish up at the bottom of the uprights. What do you notice? Can you prove your observation?

Amida is used for drawing lots and it works equally well for any number of players. The judge draws a framework on a piece of paper with one upright for each player and marks 'winning positions' at the foot of one or more of the uprights. The paper is rolled up without giving the players time to study the diagram leaving the tops of the uprights showing at the edge of the roll. Each player chooses a different upright and writes his or her name on it. The paper is then unrolled, the paths charted and the results declared. Can you prove that no two paths ever end up at the foot of the same upright?

Is it always possible for a player to win if he or she is allowed to put in just one extra rung? If you think it is not always possible draw your own framework to show this.