A378174 Maximum number of clues in a n X n crossword puzzle where each word is at least 4 letters long and the puzzle has 180-degree rotational symmetry and no interlock.

0, 0, 0, 8, 10, 12, 14, 16, 20, 36, 40, 44, 52, 64, 80, 88, 96, 108, 122, 144, 156, 166, 184, 198, 228, 244, 258, 280, 294, 332

Offset

1,4

Comments

Uses Basic American Crossword rules of: connectivity, 180-degree rotational symmetry.

Adds additional requirement of answer length at least 4.

a(1)-a(30) computed by using integer linear programming.

Because each row or column can have at most (n+1)/5 clues (consider appending a black square, and note that every clue requires 5 squares), we have a(n) <= 2n floor((n+1)/5).

Links

Table of n, a(n) for n=1..30.

Abigail Schnitzer, Illustrations of solutions for n <= 30

Abigail Schnitzer, a(14)=64

The New York Times, Submit Your Crossword Puzzles to The New York Times

Formula

Except for n = 9, 10, 12, and 14, conjectured recursive formula is a(n) = a(n-5) + 4(n-4). In particular, conjectured explicit formula is a(n) = 2n floor((n+1)/5) if mod(n,5) = 3.

Example

The trivial all-white puzzle is optimal for 4 <= n <= 8. A supporting file shows a(14)=64.

Crossrefs

Cf. A243826 (maximum number of clues when minimum word length is 3).

Sequence in context: A020744 A356657 A008557 * A161425 A096171 A337252

Adjacent sequences: A378171 A378172 A378173 * A378175 A378176 A378177

Keyword

nonn,more

Author

Abigail Schnitzer, Nov 18 2024

Status

approved