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## #151 2006-01-08 10:11:34

Ricky
Moderator

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### Re: Very interesting problems..

Here are the solutions from size 32 to 35, they seem to increase as the size does.  Also note that each solution appears twice, once forward and once in the reverse order.

Size: 32
1 8 28 21 4 32 17 19 30 6 3 13 12 24 25 11 5 31 18 7 29 20 16 9 27 22 14 2 23 26 10 15
Size: 32
1 15 10 26 23 2 14 22 27 9 16 20 29 7 18 31 5 11 25 24 12 13 3 6 30 19 17 32 4 21 28 8
Size: 33
1 8 28 21 4 32 17 19 30 6 3 13 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 23 26 10 15
Size: 33
1 15 10 26 23 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 13 3 6 30 19 17 32 4 21 28 8
Size: 34
1 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24
Size: 34
1 3 13 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 23 26 10 15 34 30 6 19 17 32 4 21 28 8
Size: 34
1 3 13 12 24 25 11 14 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 34
1 3 13 12 24 25 11 14 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 34
1 3 13 23 26 10 6 19 30 34 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 4 32 17 8 28 21 15
Size: 34
1 3 13 23 26 10 6 30 19 17 32 4 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 34 15 21 28 8
Size: 34
1 3 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 6 30 34 2 14 11 25 24 12 13 23 26 10 15 21 28 8
Size: 34
1 3 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 30 6 10 26 23 13 12 24 25 11 14 2 34 15 21 28 8
Size: 34
1 3 33 31 18 7 29 20 16 9 27 22 14 2 34 30 6 19 17 32 4 5 11 25 24 12 13 23 26 10 15 21 28 8
Size: 34
1 3 33 31 18 7 29 20 16 9 27 22 14 2 34 30 19 6 10 26 23 13 12 24 25 11 5 4 32 17 8 28 21 15
Size: 34
1 8 28 21 4 32 17 19 6 30 34 15 10 26 23 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 13 3
Size: 34
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 14 11 25 24 12 13 3
Size: 34
1 8 28 21 15 10 26 23 13 12 24 25 11 5 4 32 17 19 6 30 34 2 14 22 27 9 16 20 29 7 18 31 33 3
Size: 34
1 8 28 21 15 10 26 23 13 12 24 25 11 14 2 34 30 6 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 3
Size: 34
1 8 28 21 15 34 2 14 11 25 24 12 13 23 26 10 6 30 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 3
Size: 34
1 8 28 21 15 34 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 4 32 17 19 30 6 10 26 23 13 3
Size: 34
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 14 11 25 24 12 13 3
Size: 34
1 8 28 21 15 34 30 19 17 32 4 12 13 3 6 10 26 23 2 14 22 27 9 16 33 31 18 7 29 20 5 11 25 24
Size: 34
1 15 21 28 8 17 32 4 5 11 25 24 12 13 23 26 10 6 19 30 34 2 14 22 27 9 16 20 29 7 18 31 33 3
Size: 34
1 15 21 28 8 17 32 4 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 34 30 19 6 10 26 23 13 3
Size: 34
1 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 23 26 10 6 3 13 12 4 32 17 19 30 34 15 21 28 8
Size: 34
1 24 25 11 5 20 29 7 18 31 33 16 9 27 22 14 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3
Size: 35
1 3 6 19 30 34 2 7 18 31 33 16 9 27 22 14 11 25 24 12 13 23 26 10 15 21 28 8 17 32 4 5 20 29 35
Size: 35
1 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 7 18 31 33 16 9 27 22 14 35 29 20 5 11 25 24
Size: 35
1 3 13 12 24 25 11 5 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 14 22 27 9 7 18 31 33 16 20 29 35
Size: 35
1 3 13 12 24 25 11 5 20 29 35 14 22 27 9 16 33 31 18 7 2 23 26 10 15 34 30 6 19 17 32 4 21 28 8
Size: 35
1 3 13 12 24 25 11 14 22 27 9 16 33 31 18 7 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 5 20 29 35
Size: 35
1 3 13 23 26 10 6 19 30 34 2 7 18 31 33 16 9 27 22 14 35 29 20 5 11 25 24 12 4 32 17 8 28 21 15
Size: 35
1 3 13 23 26 10 6 30 19 17 32 4 12 24 25 11 5 20 29 35 14 22 27 9 16 33 31 18 7 2 34 15 21 28 8
Size: 35
1 3 22 27 9 7 18 31 33 16 20 29 35 14 2 34 30 6 19 17 32 4 5 11 25 24 12 13 23 26 10 15 21 28 8
Size: 35
1 3 22 27 9 7 18 31 33 16 20 29 35 14 2 34 30 19 6 10 26 23 13 12 24 25 11 5 4 32 17 8 28 21 15
Size: 35
1 3 22 27 9 16 33 31 18 7 2 14 11 25 24 12 13 23 26 10 6 19 30 34 15 21 28 8 17 32 4 5 20 29 35
Size: 35
1 3 22 27 9 16 33 31 18 7 2 14 35 29 20 5 11 25 24 12 13 23 26 10 15 34 30 6 19 17 32 4 21 28 8
Size: 35
1 3 22 27 9 16 33 31 18 7 2 34 30 6 19 17 32 4 5 20 29 35 14 11 25 24 12 13 23 26 10 15 21 28 8
Size: 35
1 3 22 27 9 16 33 31 18 7 2 34 30 19 6 10 26 23 13 12 24 25 11 14 35 29 20 5 4 32 17 8 28 21 15
Size: 35
1 8 28 21 4 32 17 19 6 30 34 2 14 35 29 7 18 31 33 3 22 27 9 16 20 5 11 25 24 12 13 23 26 10 15
Size: 35
1 8 28 21 4 32 17 19 6 30 34 2 14 35 29 20 16 33 3 22 27 9 7 18 31 5 11 25 24 12 13 23 26 10 15
Size: 35
1 8 28 21 4 32 17 19 6 30 34 15 10 26 23 2 7 18 31 33 16 9 27 22 14 35 29 20 5 11 25 24 12 13 3
Size: 35
1 8 28 21 4 32 17 19 6 30 34 15 10 26 23 2 14 22 27 9 7 18 31 5 11 25 24 12 13 3 33 16 20 29 35
Size: 35
1 8 28 21 4 32 17 19 6 30 34 15 10 26 23 2 14 22 27 9 16 20 5 11 25 24 12 13 3 33 31 18 7 29 35
Size: 35
1 8 28 21 4 32 17 19 6 30 34 15 10 26 23 13 12 24 25 11 5 20 29 35 14 2 7 18 31 33 16 9 27 22 3
Size: 35
1 8 28 21 4 32 17 19 30 6 3 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 13 23 26 10 15 34 2 14 35
Size: 35
1 8 28 21 4 32 17 19 30 6 3 33 16 9 27 22 14 11 25 24 12 13 23 26 10 15 34 2 7 18 31 5 20 29 35
Size: 35
1 8 28 21 4 32 17 19 30 6 10 26 23 13 12 24 25 11 5 20 16 9 27 22 3 33 31 18 7 29 35 14 2 34 15
Size: 35
1 8 28 21 4 32 17 19 30 6 10 26 23 13 12 24 25 11 5 31 18 7 9 27 22 3 33 16 20 29 35 14 2 34 15
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 5 11 25 24 12 13 3 33 31 18 7 29 20 16 9 27 22 14 35
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 5 20 16 9 27 22 14 11 25 24 12 13 3 33 31 18 7 29 35
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 3 13 12 24 25 11 14 35
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 5 31 18 7 9 27 22 14 11 25 24 12 13 3 33 16 20 29 35
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 12 13 3 33 16 20 29 35 14 22 27 9 7 18 31 5 11 25 24
Size: 35
1 8 28 21 15 10 26 23 2 34 30 6 19 17 32 4 12 13 3 33 31 18 7 29 35 14 22 27 9 16 20 5 11 25 24
Size: 35
1 8 28 21 15 10 26 23 13 3 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 4 32 17 19 6 30 34 2 14 35
Size: 35
1 8 28 21 15 10 26 23 13 3 33 16 9 27 22 14 11 25 24 12 4 32 17 19 6 30 34 2 7 18 31 5 20 29 35
Size: 35
1 8 28 21 15 10 26 23 13 12 4 32 17 19 6 30 34 2 14 35 29 7 18 31 33 3 22 27 9 16 20 5 11 25 24
Size: 35
1 8 28 21 15 10 26 23 13 12 4 32 17 19 6 30 34 2 14 35 29 20 16 33 3 22 27 9 7 18 31 5 11 25 24
Size: 35
1 8 28 21 15 10 26 23 13 12 24 25 11 5 4 32 17 19 6 30 34 2 14 35 29 20 16 33 31 18 7 9 27 22 3
Size: 35
1 8 28 21 15 10 26 23 13 12 24 25 11 14 2 34 30 6 19 17 32 4 5 20 16 9 27 22 3 33 31 18 7 29 35
Size: 35
1 8 28 21 15 10 26 23 13 12 24 25 11 14 2 34 30 6 19 17 32 4 5 31 18 7 9 27 22 3 33 16 20 29 35
Size: 35
1 8 28 21 15 10 26 23 13 12 24 25 11 14 35 29 20 5 4 32 17 19 6 30 34 2 7 18 31 33 16 9 27 22 3
Size: 35
1 8 28 21 15 34 2 7 18 31 33 16 9 27 22 14 35 29 20 5 11 25 24 12 4 32 17 19 30 6 10 26 23 13 3
Size: 35
1 8 28 21 15 34 2 14 11 25 24 12 13 23 26 10 6 30 19 17 32 4 5 20 16 9 27 22 3 33 31 18 7 29 35
Size: 35
1 8 28 21 15 34 2 14 11 25 24 12 13 23 26 10 6 30 19 17 32 4 5 31 18 7 9 27 22 3 33 16 20 29 35
Size: 35
1 8 28 21 15 34 2 14 22 27 9 7 18 31 5 11 25 24 12 4 32 17 19 30 6 10 26 23 13 3 33 16 20 29 35
Size: 35
1 8 28 21 15 34 2 14 22 27 9 16 20 5 11 25 24 12 4 32 17 19 30 6 10 26 23 13 3 33 31 18 7 29 35
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 5 11 25 24 12 13 3 33 31 18 7 29 20 16 9 27 22 14 35
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 5 20 16 9 27 22 14 11 25 24 12 13 3 33 31 18 7 29 35
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 5 20 29 7 18 31 33 16 9 27 22 3 13 12 24 25 11 14 35
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 5 31 18 7 9 27 22 14 11 25 24 12 13 3 33 16 20 29 35
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 12 13 3 33 16 20 29 35 14 22 27 9 7 18 31 5 11 25 24
Size: 35
1 8 28 21 15 34 2 23 26 10 6 30 19 17 32 4 12 13 3 33 31 18 7 29 35 14 22 27 9 16 20 5 11 25 24
Size: 35
1 8 28 21 15 34 30 19 17 32 4 5 11 25 24 12 13 3 6 10 26 23 2 14 22 27 9 7 18 31 33 16 20 29 35
Size: 35
1 8 28 21 15 34 30 19 17 32 4 5 11 25 24 12 13 23 26 10 6 3 33 31 18 7 2 14 22 27 9 16 20 29 35
Size: 35
1 8 28 21 15 34 30 19 17 32 4 12 13 3 6 10 26 23 2 7 18 31 33 16 9 27 22 14 35 29 20 5 11 25 24
Size: 35
1 8 28 21 15 34 30 19 17 32 4 12 13 23 26 10 6 3 22 27 9 16 33 31 18 7 2 14 35 29 20 5 11 25 24
Size: 35
1 15 10 26 23 13 12 24 25 11 5 20 16 9 27 22 3 33 31 18 7 29 35 14 2 34 30 6 19 17 32 4 21 28 8
Size: 35
1 15 10 26 23 13 12 24 25 11 5 31 18 7 9 27 22 3 33 16 20 29 35 14 2 34 30 6 19 17 32 4 21 28 8
Size: 35
1 15 21 28 8 17 32 4 5 11 25 24 12 13 23 26 10 6 19 30 34 2 14 35 29 20 16 33 31 18 7 9 27 22 3
Size: 35
1 15 21 28 8 17 32 4 5 20 29 35 14 11 25 24 12 13 23 26 10 6 19 30 34 2 7 18 31 33 16 9 27 22 3
Size: 35
1 15 21 28 8 17 32 4 12 13 23 26 10 6 19 30 34 2 14 35 29 7 18 31 33 3 22 27 9 16 20 5 11 25 24
Size: 35
1 15 21 28 8 17 32 4 12 13 23 26 10 6 19 30 34 2 14 35 29 20 16 33 3 22 27 9 7 18 31 5 11 25 24
Size: 35
1 15 21 28 8 17 32 4 12 24 25 11 5 20 16 9 27 22 14 2 34 30 19 6 10 26 23 13 3 33 31 18 7 29 35
Size: 35
1 15 21 28 8 17 32 4 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 3 13 23 26 10 6 19 30 34 2 14 35
Size: 35
1 15 21 28 8 17 32 4 12 24 25 11 5 20 29 35 14 22 27 9 16 33 31 18 7 2 34 30 19 6 10 26 23 13 3
Size: 35
1 15 21 28 8 17 32 4 12 24 25 11 5 31 18 7 9 27 22 14 2 34 30 19 6 10 26 23 13 3 33 16 20 29 35
Size: 35
1 15 21 28 8 17 32 4 12 24 25 11 14 22 27 9 16 33 3 13 23 26 10 6 19 30 34 2 7 18 31 5 20 29 35
Size: 35
1 15 34 2 14 35 29 7 18 31 33 3 22 27 9 16 20 5 11 25 24 12 13 23 26 10 6 30 19 17 32 4 21 28 8
Size: 35
1 15 34 2 14 35 29 20 16 33 3 22 27 9 7 18 31 5 11 25 24 12 13 23 26 10 6 30 19 17 32 4 21 28 8
Size: 35
1 24 25 11 5 20 16 9 27 22 3 33 31 18 7 29 35 14 2 34 30 6 19 17 32 4 12 13 23 26 10 15 21 28 8
Size: 35
1 24 25 11 5 20 16 9 27 22 3 33 31 18 7 29 35 14 2 34 30 19 6 10 26 23 13 12 4 32 17 8 28 21 15
Size: 35
1 24 25 11 5 20 16 9 27 22 14 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3 33 31 18 7 29 35
Size: 35
1 24 25 11 5 20 16 9 27 22 14 35 29 7 18 31 33 3 13 12 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 24 25 11 5 20 16 9 27 22 14 35 29 7 18 31 33 3 13 12 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 24 25 11 5 20 29 7 18 31 33 16 9 27 22 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 14 35
Size: 35
1 24 25 11 5 20 29 35 14 2 7 18 31 33 16 9 27 22 3 6 10 26 23 13 12 4 32 17 19 30 34 15 21 28 8
Size: 35
1 24 25 11 5 20 29 35 14 22 27 9 16 33 31 18 7 2 23 26 10 6 3 13 12 4 32 17 19 30 34 15 21 28 8
Size: 35
1 24 25 11 5 20 29 35 14 22 27 9 16 33 31 18 7 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3
Size: 35
1 24 25 11 5 31 18 7 9 27 22 3 33 16 20 29 35 14 2 34 30 6 19 17 32 4 12 13 23 26 10 15 21 28 8
Size: 35
1 24 25 11 5 31 18 7 9 27 22 3 33 16 20 29 35 14 2 34 30 19 6 10 26 23 13 12 4 32 17 8 28 21 15
Size: 35
1 24 25 11 5 31 18 7 9 27 22 14 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3 33 16 20 29 35
Size: 35
1 24 25 11 5 31 18 7 9 27 22 14 35 29 20 16 33 3 13 12 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 24 25 11 5 31 18 7 9 27 22 14 35 29 20 16 33 3 13 12 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 24 25 11 14 22 27 9 16 33 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 7 18 31 5 20 29 35
Size: 35
1 35 14 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3 22 27 9 16 33 31 18 7 29 20 5 11 25 24
Size: 35
1 35 14 2 34 15 10 26 23 13 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 3 6 30 19 17 32 4 21 28 8
Size: 35
1 35 14 2 34 30 6 19 17 32 4 12 24 25 11 5 20 29 7 18 31 33 16 9 27 22 3 13 23 26 10 15 21 28 8
Size: 35
1 35 14 2 34 30 19 6 10 26 23 13 3 22 27 9 16 33 31 18 7 29 20 5 11 25 24 12 4 32 17 8 28 21 15
Size: 35
1 35 14 11 25 24 12 13 3 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 35 14 11 25 24 12 13 3 22 27 9 16 33 31 18 7 29 20 5 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 35 14 22 27 9 16 20 29 7 18 31 33 3 13 12 24 25 11 5 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 35 14 22 27 9 16 20 29 7 18 31 33 3 13 12 24 25 11 5 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 35 29 7 18 31 33 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 14 22 27 9 16 20 5 11 25 24
Size: 35
1 35 29 7 18 31 33 3 13 12 24 25 11 5 20 16 9 27 22 14 2 23 26 10 15 34 30 6 19 17 32 4 21 28 8
Size: 35
1 35 29 7 18 31 33 3 13 12 24 25 11 14 22 27 9 16 20 5 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 35 29 7 18 31 33 3 13 12 24 25 11 14 22 27 9 16 20 5 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 35 29 7 18 31 33 3 13 23 26 10 6 19 30 34 2 14 22 27 9 16 20 5 11 25 24 12 4 32 17 8 28 21 15
Size: 35
1 35 29 7 18 31 33 3 13 23 26 10 6 30 19 17 32 4 12 24 25 11 5 20 16 9 27 22 14 2 34 15 21 28 8
Size: 35
1 35 29 7 18 31 33 3 22 27 9 16 20 5 4 32 17 19 6 30 34 2 14 11 25 24 12 13 23 26 10 15 21 28 8
Size: 35
1 35 29 7 18 31 33 3 22 27 9 16 20 5 4 32 17 19 30 6 10 26 23 13 12 24 25 11 14 2 34 15 21 28 8
Size: 35
1 35 29 20 5 4 32 17 8 28 21 15 10 26 23 13 12 24 25 11 14 22 27 9 16 33 31 18 7 2 34 30 19 6 3
Size: 35
1 35 29 20 5 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 7 18 31 33 16 9 27 22 14 11 25 24 12 13 3
Size: 35
1 35 29 20 5 4 32 17 8 28 21 15 34 30 19 6 10 26 23 13 12 24 25 11 14 2 7 18 31 33 16 9 27 22 3
Size: 35
1 35 29 20 5 31 18 7 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 12 13 3 33 16 9 27 22 14 11 25 24
Size: 35
1 35 29 20 5 31 18 7 2 34 15 10 26 23 13 12 24 25 11 14 22 27 9 16 33 3 6 30 19 17 32 4 21 28 8
Size: 35
1 35 29 20 5 31 18 7 2 34 30 6 19 17 32 4 12 24 25 11 14 22 27 9 16 33 3 13 23 26 10 15 21 28 8
Size: 35
1 35 29 20 5 31 18 7 2 34 30 19 6 10 26 23 13 3 33 16 9 27 22 14 11 25 24 12 4 32 17 8 28 21 15
Size: 35
1 35 29 20 16 9 27 22 14 2 7 18 31 33 3 6 10 26 23 13 12 24 25 11 5 4 32 17 19 30 34 15 21 28 8
Size: 35
1 35 29 20 16 33 3 13 12 4 32 17 8 28 21 15 34 30 19 6 10 26 23 2 14 22 27 9 7 18 31 5 11 25 24
Size: 35
1 35 29 20 16 33 3 13 12 24 25 11 5 31 18 7 9 27 22 14 2 23 26 10 15 34 30 6 19 17 32 4 21 28 8
Size: 35
1 35 29 20 16 33 3 13 12 24 25 11 14 22 27 9 7 18 31 5 4 32 17 19 6 30 34 2 23 26 10 15 21 28 8
Size: 35
1 35 29 20 16 33 3 13 12 24 25 11 14 22 27 9 7 18 31 5 4 32 17 19 30 6 10 26 23 2 34 15 21 28 8
Size: 35
1 35 29 20 16 33 3 13 23 26 10 6 19 30 34 2 14 22 27 9 7 18 31 5 11 25 24 12 4 32 17 8 28 21 15
Size: 35
1 35 29 20 16 33 3 13 23 26 10 6 30 19 17 32 4 12 24 25 11 5 31 18 7 9 27 22 14 2 34 15 21 28 8
Size: 35
1 35 29 20 16 33 3 22 27 9 7 18 31 5 4 32 17 19 6 30 34 2 14 11 25 24 12 13 23 26 10 15 21 28 8
Size: 35
1 35 29 20 16 33 3 22 27 9 7 18 31 5 4 32 17 19 30 6 10 26 23 13 12 24 25 11 14 2 34 15 21 28 8
Size: 35
1 35 29 20 16 33 31 18 7 9 27 22 14 2 23 26 10 6 3 13 12 24 25 11 5 4 32 17 19 30 34 15 21 28 8
Size: 35
1 35 29 20 16 33 31 18 7 9 27 22 14 2 23 26 10 6 19 30 34 15 21 28 8 17 32 4 5 11 25 24 12 13 3

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #152 2006-01-08 21:32:10

seerj
Member

Offline

### Re: Very interesting problems..

NICE WORK GUY!!

Can u find a list greater than 59?

## #153 2006-01-09 07:14:25

krassi_holmz
Real Member

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### Re: Very interesting problems..

I've developed another algoritm. It generates all square chains and checks if they're circular.
Size 36:
{1, 3, 6, 19, 30, 34, 2, 23, 26, 10, 15, 21, 4, 32, 17, 8, 28, 36, 13, 12, 24, 25, 11, 5, 20, 29, 7, 18, 31, 33, 16, 9, 27, 22, 14, 35}
~5min.
It seems that for all n > 32 there exist a circular sequence. Strange.

IPBLE:  Increasing Performance By Lowering Expectations.

## #154 2006-01-09 08:32:11

seerj
Member

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### Re: Very interesting problems..

#### krassi_holmz wrote:

I've developed another algoritm. It generates all square chains and checks if they're circular.
Size 36:
{1, 3, 6, 19, 30, 34, 2, 23, 26, 10, 15, 21, 4, 32, 17, 8, 28, 36, 13, 12, 24, 25, 11, 5, 20, 29, 7, 18, 31, 33, 16, 9, 27, 22, 14, 35}
~5min.
It seems that for all n > 32 there exist a circular sequence. Strange.

Very good Krassi, thanks.
Anyone know why 32 is the minimum number?

I'm running a program made with the Ricky's algorithm but till now [it has been passed 2 hours] it reached n=41..

however THANK U ALL FRIENDS!!

## #155 2006-01-09 20:17:27

seerj
Member

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### Re: Very interesting problems..

O my.. i let my program running all night long but it  crashed at size 43!!!!!! :\\\

this is the winner of the contest:
size=61

1,3,6,10,15,21,4,5,11,14,50,31,18,7,2,23,58,42,22,59,41,8,28,53,47,34, 30,51,49,32,17,19,45,36,13,12,37,27,54,46,35,29,52,48,33,16,20,44,56, 25,39,61,60,40,9,55,26,38,43,57,24

Last edited by seerj (2006-01-09 20:19:49)

## #156 2006-01-10 02:23:52

Ricky
Moderator

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### Re: Very interesting problems..

O my.. i let my program running all night long but it  crashed at size 43!!!!!! :\\\

Crashed?  What language did you program it in?

I should have size 62 by the next time I post.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #157 2006-01-10 04:38:12

seerj
Member

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### Re: Very interesting problems..

#### Ricky wrote:

O my.. i let my program running all night long but it  crashed at size 43!!!!!! :\\\

Crashed?  What language did you program it in?

I should have size 62 by the next time I post.

I used visual c++ 6.
The contest is ended. However i say thank u!
Now it's hard to explain why 32 is the minimum number..but u can find an explanation from
http://www.geocities.com/dharwadker/hamilton/main.html  with the Hamilton Algorithm..

Look at here:
http://www.mat.uniroma2.it/~tauraso/Natale/congettura.txt
is made by my teacher.

Last edited by seerj (2006-01-10 04:39:12)

## #158 2006-01-11 03:31:05

Ricky
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### Re: Very interesting problems..

Would you mind posting your code?  I'm kind of curious as to why it bombed out.

As for a list of 62:

1 3 6 10 15 21 4 5 11 14 50 31 18 7 2 34 30 19 45 36 13 51 49 32 17 47 53 28 8 41 40 60 61 20 29 52 48 33 16 9 55 26 23 58 42 39 25 56 44 37 12 24 57 43 38 62 59 22 27 54 46 35

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #159 2006-01-11 04:48:12

Ricky
Moderator

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### Re: Very interesting problems..

For every circle of size n, you must be able to contruct a Hamilton circuit.  So arrange the numbers around in a cirlce, and then draw a line for every two numbers that add up to a perfect square.

To be a ciruit, every number has to have at least two lines connected to it.  It can be observed that 2 won't have two lines going in until 14:

2 + x = perfect square, where x is a natural number not equal to two.  The lowest two perfect squares are 9 and 16 (2 and 1 don't count since x is natural and not 2).  So to get two lines connected to 2:

2 + x = 16, x = 14.

Size 14 is a minimum, when only considering 2.

For n ≥ 14, consider 8.

8 + x = perfect square

The two lowest perfect squares are 9 and 25.

8 + x = 25, x = 17

So size 17 is a minimum.

For n ≥ 17, consider 16.

16 + x = perfect square

The two lowest perfect squares are 25 and 36.

16 + x = 36, x = 20.

This pattern will continue till 32, where every natural number up to and including 32 has a least two lines connected to it.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #160 2006-01-11 05:45:55

seerj
Member

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### Re: Very interesting problems..

this is your code. I tried with it but after 6 hours it has crashed

#include <iostream>
#include <fstream>
#include <vector>
#include <math.h>
#include <windows.h>

using namespace std;

void testVals(vector<int> base, vector<bool> use, vector<int> test);
bool compare(int x, int y);

int main()
{
int start = GetTickCount();
for (int size = 2; size <= 32; size++)
{
cout << "Testing list size " << size << "..." << endl;
vector<int> base;
vector<bool> used;
vector<int> test;
for (int gen = 1; gen <= size; gen++)
{
base.push_back(gen);
used.push_back(false);
}
testVals(base, used, test);
}
int end = GetTickCount();
cout << "Time: " << (end - start) / 1000.0 << endl;
return 0;
}

void testVals(vector<int> base, vector<bool> used, vector<int> test) {
if (test.size() == base.size())
{
bool result = compare(test[test.size()-1], test[0]);
if (result)
{
ofstream o("asdf.txt", ios::app);
o << "Size: " << test.size() << endl;
for (int x = 0; x < test.size(); x++)
{
o << test[x] << " ";
}
o << endl;
}
return;
}

for (int x = 0; x < used.size(); x++)
{
if (used[x] == false)
{
if (x != 0)
{
if (!compare(test[test.size()-1], base[x])) {
continue;
}
}
used[x] = true;
test.push_back(base[x]);
testVals(base, used, test);
test.erase(test.end()-1);
used[x] = false;
}
}
}

bool compare(int x, int y)
{
double temp = sqrt((double)x + y);
const double amount = 0.00001;
if (temp + amount > (int) temp && temp - amount < (int)temp)
{
return true;
}
return false;
}

## #161 2006-01-11 07:11:52

Ricky
Moderator

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### Re: Very interesting problems..

Quite odd, mine works completely fine, even overnight.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #162 2006-01-16 00:45:11

krassi_holmz
Real Member

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### Re: Very interesting problems..

Good work!

I just want to continue the Ricky's mind.

Let n>20. Take 18.
But 36-18=18 so
18+x=49; x=31!
So n>31.

But I was wondering if there exist different proof without guessing.

IPBLE:  Increasing Performance By Lowering Expectations.

## #163 2006-01-16 00:53:46

Ricky
Moderator

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### Re: Very interesting problems..

Boy, I thought this thread was finally dead...

But I was wondering if there exist different proof without guessing.

I think there is, if you know enough about Hamiltonian Circuits.  I don't.  But I wouldn't quite call it guessing, just making observations.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #164 2006-01-16 02:59:54

krassi_holmz
Real Member

Offline

### Re: Very interesting problems..

We can generalize the question:
Let have the set {a_i} elem N. What are the conditions to exist sircular sequence {b_(1..n)} (the sum of every two consecutive members of {b} is element of {a}.

IPBLE:  Increasing Performance By Lowering Expectations.

## #165 2006-01-16 03:10:35

krassi_holmz
Real Member

Offline

### Re: Very interesting problems..

Here's nessesery condition:
There must exist i0 element of N : for every i>=i0 a_(i+2)-a_i<i.

With that we can proof that there doesn't exist cicular sequence for which the sums are Fibonacci numbers. (n>2)
if n=2 we get
1,2 It has sum 3 which is fibonacci number.

Now we'll proof that F(i+2)-F(i)>i, i>2;
F(i+2)=F(i+1)+F(i)>2F(i);
So F(i+2)-F(i)>2F(i)-F(i)=F(i)
And F(i)>i for i>2 so there doesn't exist fibonacci sircular sequence with lengh >2.

IPBLE:  Increasing Performance By Lowering Expectations.

## #166 2006-01-16 04:53:59

Ricky
Moderator

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### Re: Very interesting problems..

There must exist i0 element of N : for every i>=i0 a_(i+2)-a_i<i.

if n=2 we get

F(i+2)-F(i)>i, i>2;

You haven't defined N, a_, or F() or n.  I have no idea what these are.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

## #167 2006-01-18 03:25:03

Ricky
Moderator

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### Re: Very interesting problems..

Got my program down to 0.371 seconds when doing 2-32 by calculating where the next perfect root will be instead of linearly searching for it.

Last edited by Ricky (2006-01-18 03:25:54)

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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