|
|
A243826
|
|
Maximum number of clues in a certain class of n X n crossword puzzles.
|
|
1
|
|
|
0, 0, 6, 8, 10, 12, 22, 28, 32, 40, 50, 64, 72, 84, 96, 116, 126, 144, 158, 184, 198, 220, 236, 268, 284, 312, 332, 368, 388, 420, 442, 484, 506, 544, 570, 616, 642, 684, 712, 764, 792, 840, 872, 928, 960, 1012, 1046, 1108, 1142, 1200
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Uses New York Times rules of: connectivity, 180-degree rotational symmetry, answer length at least 3.
a(1)-a(50) computed by using integer linear programming.
Because each row or column can have at most (n+1)/4 clues (consider appending a black square, and note that every clue requires 4 squares), we have a(n) <= 2n floor((n+1)/4).
|
|
LINKS
|
|
|
FORMULA
|
Except for n = 7, 11, and 19, conjectured recursive formula is a(n) = a(n-4) + 4(n-3) - [2 if mod(n,8) in {1,7}]. In particular, conjectured explicit formula is a(n) = 2n floor((n+1)/4) if mod(n,4) = 2.
|
|
EXAMPLE
|
The trivial all-white puzzle is optimal for 3 <= n <= 6.
The Ferland paper shows that a(15) = 96.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Upper bound and conjectured formulas from Rob Pratt, Jun 23 2014
|
|
STATUS
|
approved
|
|
|
|