Maybe we can find a pattern if we play around with fewer cells? On a 1x6 strip, Kolyas' first move involves coloring two cells where the possible distances are
Distance 0: adjacent cells, eg. [imath]d(1,2),d(2,3)[/imath] etc.
Distance 1: one cell apart, [imath]d(1,3), d(2,4)[/imath], ...
Distance 2 Two cells apart [imath]d(1,4), d(2,5)[/imath] ...
Distance 3 Three cells apart [imath]d(1,5), d(2,6)[/imath] ...
Distance 4 Four cells apart [imath]d(1,6)[/imath].
I found by trial and error that Tolya can always block Kolya's ability to use unique distances in a 1 × 6 strip, no matter what Kolya picks for his first move. Let K denote Kolya and T denote Tolya, and order the moves in ascending order.
Case 1:
[math]K: d(1,2)=0 \\ T: d(3,5)=1 \\ K: d(4,6)=1[/math]
Case 2:
[math]K: d(1,3)=1\\ T: d(2,5)=2 \\ K: d(4,6)=1[/math]Case 3:
[math]K: d(1,4)=2 \\ T: d(2,5) =2 \\ K: d(3,6)=2[/math]Case 4:
[math]K: d(1,5)=3 \\T:d(3,4)=0 \\K: d(2,6)=3[/math]Case 5:
[math]K: d(1,6)=4 \\ T:d(3,5)=1 \\ K : d(2,4)=1[/math]
This reasoning assumes that a cell cannot be colored more than once.
