A353071 Maximum number of clicks needed to solve any solvable Lights Out problem on an n X n grid.

1, 4, 9, 7, 15, 36, 49, 64, 37, 100, 65, 144, 169, 123, 225, 124, 199, 324, 197, 400, 441, 484

Offset: 1

Author

William Boyles, Apr 21 2022

Keywords

• nonn
• more
• hard

Comments

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

a(n) >= A075464(n).

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.

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

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

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.

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

239 <= a(23) <= 305.

Links

• William Boyles, Most Clicks Problem in Lights Out, arXiv:2201.03452 [math.CO], 2022.

Crossrefs

Cf. A159257, A075464.