A351516 a(n) is the maximum number of 3-letter words that can be contained in an n X n crossword puzzle.

0, 0, 6, 6, 8, 12, 20, 24, 30, 36, 48

Offset: 1

Author

Rodolfo Kurchan, Feb 12 2022

Keywords

• nonn
• more

Comments

The n X n crossword puzzle grid must contain the maximum number of 3-letter words (and no words of other lengths), and all words must be interconnected (i.e., no word or set of connected words can be completely separated from the other words).

The magazine Humor y Juegos Number 13 says that Mariano Pablo Amieva and Solomon Golomb found a general answer to this problem.

Examples

			4 X 4 solution with a(4) = 6 words by Jaime Poniachik:
  +---+---+---+---+
  |   |   |   |XXX|
  +---+---+---+---+
  |XXX|   |   |   |
  +---+---+---+---+
  |XXX|   |   |   |
  +---+---+---+---+
  |XXX|XXX|XXX|   |
  +---+---+---+---+
.
5 X 5 solution with a(5) = 8 words by Jaime Poniachik:
  +---+---+---+---+---+
  |   |   |   |XXX|XXX|
  +---+---+---+---+---+
  |   |XXX|   |   |   |
  +---+---+---+---+---+
  |   |   |   |XXX|   |
  +---+---+---+---+---+
  |XXX|   |XXX|XXX|   |
  +---+---+---+---+---+
  |XXX|   |   |   |XXX|
  +---+---+---+---+---+
.
6 X 6 solution with a(6) = 12 words by Jaime Poniachik:
  +---+---+---+---+---+---+
  |   |   |   |XXX|XXX|   |
  +---+---+---+---+---+---+
  |XXX|   |XXX|   |   |   |
  +---+---+---+---+---+---+
  |XXX|   |   |   |XXX|   |
  +---+---+---+---+---+---+
  |   |XXX|   |   |   |XXX|
  +---+---+---+---+---+---+
  |   |   |   |XXX|   |XXX|
  +---+---+---+---+---+---+
  |   |XXX|XXX|   |   |   |
  +---+---+---+---+---+---+
.
Example of an invalid solution: a 6 X 6 solution with a(6) = 12 words, but not valid because some words have 2 letters instead of 3, and the 3-letter words are not all interconnected (black squares completely separate the set of 3-letter words at the upper right from the set of 3-letter words at the lower left):
  +---+---+---+---+---+---+
  |   |   |XXX|   |   |   |
  +---+---+---+---+---+---+
  |   |   |XXX|   |   |   |
  +---+---+---+---+---+---+
  |XXX|XXX|XXX|   |   |   |
  +---+---+---+---+---+---+
  |   |   |   |XXX|XXX|XXX|
  +---+---+---+---+---+---+
  |   |   |   |XXX|   |   |
  +---+---+---+---+---+---+
  |   |   |   |XXX|   |   |
  +---+---+---+---+---+---+
.
		

References

• Jaime Poniachik, Las matemáticas del crucigrama, Revista Humor y Juegos, Número 9, ~ 1981.

Links

• Rodolfo Kurchan, Problem 781, Puzzle Fun 35, December 2021.