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Maximum number of clicks needed to solve any solvable Lights Out problem on an n X n grid.
0

%I #16 May 13 2022 18:13:34

%S 1,4,9,7,15,36,49,64,37,100,65,144,169,123,225,124,199,324,197,400,

%T 441,484

%N Maximum number of clicks needed to solve any solvable Lights Out problem on an n X n grid.

%C a(n) = n^2 if and only if A159257(n) = 0.

%C a(n) >= A075464(n).

%C If n = 6k-1 for some integer k, then a(n) <= 26k^2 - 12k + 1. This upper bound is equal to a(n) when A159257(n) = 2. Further, it is conjectured that if A159257(n) = 2, then n = 6k-1 for some integer k.

%C It is conjectured that if A159257(n) = 4, then n = 5k-1 for some integer k, and a(n) = 17k^2 - 10k.

%C It is conjectured that if A159257(n) = 6, then n = 12k-1 for some integer k, and a(n) = 88k^2 - 24k + 1

%C It is conjectured that if A159257(n) = 8, then either n = 10k-1 or n = 17k-1 for some integer k. If n = 10k-1, then a(n) = 60k^2 - 20k - 3. If n = 17k-1, then a(n) = 161k^2 - 34k - 3.

%C It is conjectured that if A159257(n) = 10, then n = 30k-1 for some integer k, and a(n) = 506k^2 - 60k - 3.

%C 239 <= a(23) <= 305.

%H William Boyles, <a href="https://arxiv.org/abs/2201.03452">Most Clicks Problem in Lights Out</a>, arXiv:2201.03452 [math.CO], 2022.

%Y Cf. A159257, A075464.

%K nonn,more,hard

%O 1,2

%A _William Boyles_, Apr 21 2022