cp's OEIS Frontend

This is the peaceable queens problem (A250000) for 3 players.

For n >= 11, it seems that a(n) is simply 2n - 14. However this turns out to be false as a(18) >= 23.

In the limit of large n, Arthur O'Dwyer (see links) showed that the optimal value is lower bounded by 0.0718*n^2. All currently known best solutions follow this formula (when rounded down). - M. A. Achterberg, Dec 01 2022

One has T(n, k) = 0 exactly when (n, k) = (2, 2) or (3, 3).

One has T(n, n) = 1 except when n = 2 or 3 (that is, when A000170(n) = 0).

For a fixed k, the sequence T(-, k) is nondecreasing.

For a fixed n, the sequence T(n, -) is nonincreasing.

For a fixed nonzero p, the sequence (T(k + p, k))_k is nonincreasing. Indeed, given a configuration with k+1 armies on a k+p+1 chessboard, remove the row and column containing a given queen; this row and column can contain only queens of one army, so this yields a configuration of k armies on a k+p chessboard.

One has T(n, 1) = n^2 and T(n, 2) = A250000(n).

For a fixed k, one has, asymptotically in n, that 1/2.(n/k)^2 <= T(n, k) <= (n/k)^2, which can be proved as follows.

For the upper bound, let R(n, k) be defined as T(n, k) with rooks instead of queens. Then, T(n, k) <= R(n, k) ~ (n/k)^2. Indeed, for 1 <= i <= k, say the i^th army controls a_i rows and b_i columns. The sum of the a_i's, as well as the sum of the b_i's, is at most n. The size of the i^th army is at most a_i b_i. Therefore, one wants to maximize min(a_i b_i, i = 1..k). Ignoring rounding, the maximum is reached when all the a_i's and b_i's are equal.

For the lower bound, consider the n X n chessboard as a k X k grid with cells of size (n/k) X (n/k). Consider a configuration of k non-attacking queens on the k X k chessboard as in A000170(k). Place each of the k armies inside one cell occupied in that configuration. The precise placement is as follows: the army occupies the square whose vertices are the midpoints of the sides of the cell, hence has size 1/2.(n/k)^2. These armies are non-attacking.