Maximizing Crossword Grids with a Fixed Minimum Clue Length

Abstract

In an American-style crossword puzzle, the grid must be connected, 180-degree symmetric, and have minimum word length 3. Ferland and Pratt construct $n\times n$ crossword grids with many clues satisfying these conditions and prove their construction is maximal when $n\equiv 1$ or 3 $\pmod{4}$. In this paper, we explore American-style crosswords in which the minimum word length is 4. Jacobs computed the maximal $15 \times 15$ grid with minimum word length 4. In this paper, we construct grids with a maximal number of clues for all $n \not \equiv 0 \pmod{5}$.