85. Contest. Euclid Paracelsus Bombast Umbugio tries to supplement his meager professor's salary by participating in company-sponsored soap production contests. In one such competition, it was required to determine the number of paths along which the word MATHEMATICIAN* could be read on a given diagram. Umbugio counted 1587 paths starting from one of the first five lines. When it was time to announce the results, he was quite perplexed, to say the least. Help the professor count all the paths using as few calculations as possible.
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M
                                    M A M
M A T A M
M A T H T A M
M A T H E H T A M
M A T H E M E H T A M
M A T H E M A M E H T A M
M A T H E M A T A M E H T A M
M A T H E M A T I T A M E H T A M
M A T H E M A T I C I T A M E H T A M
M A T H E M A T I C I C I T A M E H T T A M
M A T H E M A T I C I A I C I T A M E H T A M
M A T H E M A T I C I A N A I C I T A M E E H T A A M
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85. A path can be traced by moving "backwards" from $N$. If we consider the left half of the diagram, including the central column, then at each step backward we have a choice between two possible directions, giving us $2^{12}$ paths. Doubling this number and subtracting 1 (to avoid counting the central column twice), we get $2^{13}-1=8191$ paths.

[J. F. Leech, A. M. M., 68, 296 (March 1961).]