Sunday, 28 September 2014

Most Asymmetric Tour?

How does one assess how asymmetric a tour is? Or indeed any object? Is there an objective measure of the degree of asymmetry?

Thinking over this question the last two days I have come up with the tour shown which I mantain is the most asymmetric tour possible. There may be better, or alternative ways of measuring asymmetry, but I have concluded that in the case of a tour it is the number of the 21 geometrically distinct moves that occur an odd number of times. In this tour the total is 18, and moreover 12 of these occur only once.

The only moves that occur an even number of times are the corner moves like a1-c2 (where there are eight, which occur in every closed tour) the edge-to edge moves of the type a2-c1 (eight again), and the pair of moves a3-c2 and h3-f2 which are of the same type. Two moves are considered to be of the same type if one can be put in the position of the other by a rotation or reflection of the board.

The 12 moves that occur only once are b1-c3, d1-c3, f1-e3, b2-c4, d2-b3, d2-f3, d3-c5, e3-d5, a4-b6, e4-g5, b5-d6, e5-f7.