Enigmatics |
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Posted on: Sunday 9/12, 2007; 12:40 PM Enumerate all of the possible states {set,harry,tom} that the match can reach.
Define a function that computes whether an element of the adjacency matrix for transitions between states is True or False, where each (row,column) entry corresponds to an (input,output) state pair. This function is True for allowed ways of winning a point within a set or for allowed ways in which set 1 can be in a winning state followed by starting set 2. Compute the adjacency matrix (consisting of 1s for allowed transitions, and 0s elsewhere) corresponding to the allowed transitions above.
Compute all non-zero iterations of the adjacency matrix to determine what states are reachable after each possible number of games. Display the iterated adjacency matrices. This sequence needs to go only as far as the 27th game, because this exhausts all possible ways of getting to the end of set 2 . In each matrix a (row,column) entry is the number of ways of making the transition from an input state (row) to a output state (column). The match starts in state 1, corresponding to the first row of each iterated adjacency matrix above.
Keep only the set 2 part of the first row of each iterated adjacency matrix, and clip each matrix element to 0 or 1 to indicate only the presence or absence of paths to the output state. Extract the states reachable in set 2. Extract the cases in set 2 where new balls are needed (i.e. after 7, 7+9, and 7+9+9 games), and convert the 1's to states {set,harry,tom}. Detect which states are missing. These are the scores at which Dick can make his mis-calls for new balls. Compute all the triples of miscall states, and then extract only those cases that are legal. There is only one case that survives, which is thus the solution. |
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