Source: Based on a problem from The Little Giant Encyclopedia of Puzzles.
From Al Zimmermann, for the 24 solutions for N=10:
1, 4, 5, 8, 9, 2, 3, 6, 7, 10 1, 4, 5, 10, 7, 2, 3, 6, 9, 8 1, 4, 7, 6, 9, 2, 3, 8, 5, 10 1, 4, 7, 10, 5, 2, 3, 8, 9, 6 1, 4, 9, 6, 7, 2, 3, 10, 5, 8 1, 4, 9, 8, 5, 2, 3, 10, 7, 6 1, 6, 3, 8, 9, 2, 5, 4, 7, 10 1, 6, 3, 10, 7, 2, 5, 4, 9, 8 1, 6, 7, 4, 9, 2, 5, 8, 3, 10 1, 6, 9, 4, 7, 2, 5, 10, 3, 8 1, 7, 3, 9, 5, 6, 2, 8, 4, 10 1, 7, 3, 10, 4, 6, 2, 8, 5, 9 1, 7, 4, 8, 5, 6, 2, 9, 3, 10 1, 7, 4, 10, 3, 6, 2, 9, 5, 8 1, 7, 5, 8, 4, 6, 2, 10, 3, 9 1, 7, 5, 9, 3, 6, 2, 10, 4, 8 1, 8, 2, 9, 5, 6, 3, 7, 4, 10 1, 8, 2, 10, 4, 6, 3, 7, 5, 9 1, 8, 3, 6, 9, 2, 7, 4, 5, 10 1, 8, 4, 7, 5, 6, 3, 9, 2, 10 1, 8, 5, 4, 9, 2, 7, 6, 3, 10 1, 8, 5, 7, 4, 6, 3, 10, 2, 9 1, 9, 2, 8, 5, 6, 4, 7, 3, 10 1, 9, 3, 7, 5, 6, 4, 8, 2, 10For N=6:
1, 4, 5, 2, 3, 6 1, 5, 3, 4, 2, 6
Going around clockwise: 1, 10, 2, 8, 4, 6, 5, 7, 3, 9 I could find only one assymetric solution. Cannot be done with 1 to 8 (in general powers of 4). Can be done with 6 (powers of 2 that are not powers of 4). 1, 6, 2, 4, 3, 5 The algorithm to get this soln is very simple. 1) start with 1, place n/2 + 1 opposite it. Now perform in loop: 2) pick the next 2 biggest (initially n, n-1) and put it next to two smallest points (initially next to 1). Fill their opposite points. 3) pick the next 2 smallest and put it next to the two biggest points. Fill their opposite points. For eg. in case of n = 14, you get: Alg step: Locations: Numbers: 1 1, 8 1, 8 2.1 2, 9, 14, 7 14, 7, 13, 6 3.1 3, 10, 13, 6 2, 9, 3, 10 2.2 4, 11, 12, 5 12, 5, 11, 4 Done.