[Retros] Ortho-reconstructions

[Mario Richter]

Dear Nikolai,

still studying your Ortho-reconstructions paper I found a rather
surprising cook in your problem N14: the first number k of single
moves, after which the starting position is again on the board,
is indeed 40.
But the next k with that property is not, as expected, 80, but 76:

1.Qg1-h2 Rf1-g1 2.Rf2-f1 Re2-f2 3.Rd2-e2 Rd1-d2 4.Rc1-d1 Qb1-c1 
5.Ra1-b1 Bb2-a1 6.Rb1-b2 Qc1-b1 7.Rd1-c1 Rd2-d1 8.Re2-d2 Rf2-e2 
9.Rf1-f2 Rg1-f1 10.Qh2-g1 Rh1-h2 11.Qg1-h1 Rf1-g1 12.Rf2-f1 Re2-f2 
13.Rd2-e2 Rd1-d2 14.Rc1-d1 Qb1-c1 15.Rb2-b1 Qc1-b2 16.Rd1-c1 Rd2-d1 
17.Re2-d2 Rf2-e2 18.Rf1-f2 Rg1-f1 19.Qh1-g1 Rh2-h1 20.Qg1-h2 Rf1-g1 
21.Rf2-f1 Re2-f2 22.Rd2-e2 Rd1-d2 23.Rc1-d1 Qb2-c1 24.Rb1-b2 Qc1-b1 
25.Rd1-c1 Rd2-d1 26.Re2-d2 Rf2-e2 27.Rf1-f2 Rg1-f1 28.Qh2-g1 Rh1-h2 
29.Qg1-h1 Rf1-g1 30.Rf2-f1 Re2-f2 31.Rd2-e2 Rd1-d2 32.Rc1-d1 Qb1-c1 
33.Rb2-b1 Ba1-b2 34.Rb1-a1 Qc1-b1 35.Rd1-c1 Rd2-d1 36.Re2-d2 Rf2-e2 
37.Rf1-f2 Rg1-f1 38.Qh1-g1 Rh2-h1

This leads to an interesting question:
A position can be repeated after k, 2k, 3k, ..., i*k single moves.
After what value of i can one be sure that the position has indeed the
"cyclic repetion property"?

Best regards,

  mario