I could not think of a way to solve this riddle by hand (besides the observations already made by Rand al'Thor.
So, I wrote a small program to solve this riddle. The key to get acceptable run-time was to implement some flood-fill algorithm to check that the current chain does not split the board in two disconnected region. This is easy to spot for humans, but not for computers.
Here is the solution:
57 58 [ 67] 68 69 76 77 80 81 82 56 [ 59] 66 65 70 75 78 [ 79] 84 [ 83] 55 60 [ 61] 64 [ 71] 74 [ 31] 32 85 86 54 [ 53] 62 63 72 [ 73] 30 33 100 87 51 52 15 14 [ 13] 12 [ 29] 34 99 88 50 [ 17] 16 1 [ 2] [ 11] 28 35 98 [ 89] 49 18 [ 5] 4 [ 3] 10 27 36 [ 97] 90 48 [ 19] 6 [ 7] 8 9 26 [ 37] 96 91 [ 47] 20 21 22 [ 23] 24 25 38 95 92 46 45 44 [ 43] 42 [ 41] 40 39 94 93
In total 8 solutions were found. The numbers 86-100 can be reversed in all of them reducing the total to 4. Two of these are obtained by reversing 8-16 and 8-22 of the solution above. Here is a fourth type, I do not see how this can be obtained from the first:
57 58 [ 67] 68 69 76 77 80 81 82 56 [ 59] 66 65 70 75 78 [ 79] 84 [ 83] 55 60 [ 61] 64 [ 71] 74 [ 29] 28 85 100 54 [ 53] 62 63 72 [ 73] 30 27 86 99 51 52 45 44 [ 43] 42 [ 31] 26 87 98 50 [ 47] 46 1 [ 2] [ 41] 32 25 88 [ 97] 49 48 [ 5] 4 [ 3] 40 33 24 [ 89] 96 12 [ 11] 6 [ 7] 38 39 34 [ 23] 90 95 [ 13] 10 9 8 [ 37] 36 35 22 91 94 14 15 16 [ 17] 18 [ 19] 20 21 92 93
And here is a grid of all numbers that are the same in all solutions.
57 58 [ 67] 68 69 76 77 80 81 82 56 [ 59] 66 65 70 75 78 [ 79] 84 [ 83] 55 60 [ 61] 64 [ 71] 74 [...] ... 85 ... 54 [ 53] 62 63 72 [ 73] 30 ... ... ... 51 52 ... ... [...] ... [...] ... ... ... 50 [...] ... 1 [ 2] [...] ... ... ... [...] 49 ... [ 5] 4 [ 3] ... ... ... [...] ... ... [...] ... [...] ... ... ... [...] ... ... [...] ... ... ... [...] ... ... ... ... ... ... ... ... [...] ... [...] ... ... ... 93
