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For fixed $n$, you can solve the problem via integer linear programming as follows. Let $C_k$ be the set of cells in clique $k$ (row, column, or wrapped diagonal). … The problem is to maximize $\sum_i \sum_j x_{ij}$ subject to linear constraints $$\sum_{(i,j)\in C_k} x_{ij} \le 1 \quad \text{for all $k$}$$ For $n=8$, the maximum turns out to be For general $n$, …

You can solve the problem via integer linear programming as follows. Let $p_{ij}$ be the priority for assigning object $i$ to cell $j$. …

You can solve the problem via integer linear programming as follows. Let $S=[5] \times [5]$ be the set of squares. … It turns out that the linear programming relaxation (obtained by relaxing the decision variables as $x_{ij} \ge 0$ and $y_{ij\bar{i}\bar{j}} \ge 0$) yields the same optimal objective value, and the corresponding …

Via integer linear programming (specifically, set covering), here are the minimum numbers of (YAWL) words to cover every letter at least once: 12: 13: 14: 15: …

Via integer linear programming, the maximum number of unique letters is There are $520$ such solutions. If you disallow the repeated letter to be a vowel, there are …

Via integer linear programming, the unique solution is …

Via integer linear programming (https://puzzling.stackexchange.com/a/123475/65277), here are the largest numbers of covered letters. 3 3 5 5 5 5 6 6 7 7 5 5 8 8 4 4 9 9 …

Via integer linear programming, using the formulation described in my answer https://puzzling.stackexchange.com/a/128031/65277: …

You can solve the problem via integer linear programming as follows, with binary decision variables $x_{ijk}$ to indicate whether cell $(i,j)$ contains digit $k$: For each region (row, column, or $2\times4 …

Via integer linear programming (https://puzzling.stackexchange.com/a/123475/65277), I found that the maximum is There are 515 such optimal solutions, such as: …

Via integer linear programming, I found the following minimum values for $n \times n$ grids: Minimum values for $n \le 44$ are here: https://oeis.org/A365271 8x8: 9x9: 10x10: 11x11: 12x12: …

As a warm-up, here's the lexicographically minimal Sudoku, obtained via integer linear programming, with binary decision variables $x_{ijk}$ to indicate whether cell $(i,j)$ contains digit $k$: For each …

You can solve the problem via integer linear programming, with a nonnegative decision variable $x_S$ for each of the seven nonempty subsets $S$ of the three languages and linear constraints to enforce … programming relaxation also has maximum value and the corresponding dual variables provide a short certificate of optimality as follows. …

You can solve the problem via integer linear programming as follows. …

Here's one that omits only X, obtained via integer linear programming: For the 2315 common words from the list provided in What is the longest Wordle game? …