Through Mazes to Mathematics

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Where else do the maze numbers occur?

At the same time (1986-88) that I was investigating the maze numbers {M(n) = number of different S.A.T. mazes with n levels} other people in different parts of the world were thinking about the same mathematical problem.

V. I. Arnol'd

An identical counting problem was proposed by V. I. Arnol'd in Moscow; it is related to his work which appeared in the Siberian Mathematical Journal of September-October 1988. Arnol'd had defined a meander as a connected oriented non-self-intersecting curve which intersects a fixed oriented line in several points. His article contains this figure which, if tilted to the right, shows exactly the eight maze-paths of depth 6, and his M5 is the maze number M(6). Arnol'd gives the first sixteen "meandering numbers," up to M16 = M(17) = 252939.

Vladimir A. Kazakov - Ivan Kostov

At approximately the same time, the calculation of M(n) came up in another context, this time in the computation of a matrix integral arising in quantum field theory. I learned of this work from Alexander Zvonkin, whom I thank for his correspondence. Later, he and Lando published an account of this work in their 1993 paper.

Warren Smith

There was another simultaneous and completely independent occurrence of this problem. Warren Smith, working on a new, subexponential traveling salesman algorithm in his 1988 Princeton thesis defined jt(N) as the number of topologically distinct ways in which a Jordan curve can cross an oriented line in the plane at exactly 2N points, and computed up to jt(10) = M(20) = 8152860.

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Tony Phillips
Math Dept SUNY Stony Brook
tony at
March 12 1997
April 6 1999