Challenge: sum of squares. Let's break 299

[Allen]

I believe there are multiple solutions to each graph walk. For example the 999 run above returned 3 unique solutions. For the first 888 items, the graphs walks are the same, but from there on, there are slightly different cycles in the 3 found solutions.
Here are the position numbers and the values for the last 122 steps for each of the 3 solutions.

Code:


888 58   89  89
889 111 107 107
890 85 118 118
891 84 138 138
892 112 58 58
893 113 111 111
894 83 85 85
895 142 84 84
896 114 112 112
897 82 113 113
898 62 83 83
899 107 142 142
900 118 114 114
901 138 82 82
906 89 64 64
907 136 105 105
909 105 136 136
910 91 60 60
933 64 45 45
934 57 76 76
935 43 93 93
936 78 51 51
937 66 70 70
938 55 74 74
939 45 95 95
940 76 49 49
941 93 72 72
942 51 97 97
943 70 47 47
944 74 53 53
945 95 91 91
946 49 78 78
947 72 66 66
948 97 55 55
949 47 26 26
950 53 23 23
951 28 98 98
952 21 46 46
953 60 18 18
954 61 63 63
955 39 37 37
956 42 27 27
958 27 59 59
959 37 62 62
960 63 38 38
961 18 43 43
962 46 57 57
963 98 7 7
964 23 42 42
965 41 39 39
966 59 61 61
967 5 20 20
968 20 29 29
969 29 52 52
970 52 48 48
971 48 33 33
972 33 16 16
973 16 9 9
974 9 40 40
975 40 41 41
976 24 8 8
977 25 28 28
978 11 21 21
979 38 4 4
980 26 32 32
981 10 17 17
982 15 19 19
983 34 30 30
984 30 34 6
985 19 2 10
986 6 14 15
987 3 35 34
988 13 1 2
989 12 15 14
990 4 10 35
991 32 6 1
992 17 3 3
993 8 13 13
994 1 12 12
995 35 24 24
996 14 25 25
997 2 11 11
998 7 5 5

Edit to add:
Note- part of the multiple solutions has to do with "skeleton key" cycles like (6,19,30) that can appear in any order together. Mathematically these are the solutions occur when all permutations of a given numbers are subsets of the 3-node graph.

On the N=1000 graph there are only 828 such wonders (*6 if you include the permutations) out of 300300 valid combinations of 3 nodes. Each of these strings is an opportunity to create a new cycle in the graph.
Since the smallest pair of these is: (6, 19, 30) and (5, 20, 44),

If at least one solution exists for N>=44, there could be multiple solutions, since any solution could contain a pair of these special strings (one to depart the main thread and the other to come back).

A more thorough proof would prove that there are no isolated parts of the a particular graph (i.e. closed cycle), and that it contained one of these "skeleton keys".

Here are the most frequent nodes in the N=1000 graph super nodes with their associated counts:

Code:


[(2, 15),
 (8, 13),
 (20, 10),
 (30, 10),
 (32, 10),
 (38, 10),
 (44, 10),
 (92, 10),
 (12, 9),
 (14, 9),
 (18, 9),
 (42, 9),
 (48, 9),
 (58, 9),
 (68, 9),
 (78, 9),
 (80, 9),
 (132, 9),
 (194, 9),
 (248, 9)]

an interesting pattern..