OFFSET
1,3
COMMENTS
On a 4 X 4 square grid, there are 14 lattice squares parallel to the axes. What is the fewest dots you can remove from the grid such that at least one vertex of each of the 14 squares is removed? The answer is a(4) = 4. In general a(n) is the answer for an n X n grid.
This is a "set covering problem", which can be handled by integer linear programming for small n. - Robert Israel, Mar 25 2009
LINKS
Ed Wynn, A comparison of encodings for cardinality constraints in a SAT solver, arXiv:1810.12975 [cs.LO], 2018.
EXAMPLE
1 X 1: 0 dots, since there are already no squares,
2 X 2: 1 dot, any vertex will do,
3 X 3: 2 dots, the center kills all the small squares and you need one corner to kill the big square,
a(4) = 4: there are 4 disjoint squares, so it must be at least 4, and with a little more work you can find a set of 4 dots that work.
From Robert Israel: (Start)
For the 5 X 5 case, Maple confirms that the optimal solution is 8 dots,
which can be placed at
[1, 1], [1, 3], [2, 2], [2, 3], [3, 0], [3, 1], [3, 2], [4, 4]
For the 6 X 6 case, Maple tells me that the optimal solution is 12 dots,
which can be placed at
[0, 5], [1, 1], [1, 2], [1, 4], [2, 0], [2, 3], [2, 4], [3, 1], [3, 3],
[4, 0], [4, 4], [5, 2]
For the 7 X 7 case, Maple tells me that the optimal solution is 17 dots,
which can be placed at
[0, 0], [1, 2], [1, 3], [1, 5], [2, 1], [2, 4], [2, 5], [3, 2], [3, 3],
[3, 4], [4, 1], [4, 2], [4, 5], [5, 1], [5, 3], [5, 4], [6, 6]
For n=9, at least a(9) = 30 points (.) have to be removed while 51 (X) of 81 are remaining. The solution is unique (congruent images being ignored).
. X X X X X . X .
X . X . . X X X X
X X . . X . X . X
X . . X X X X . .
X X X . X . . X X
X . X X X . . . X
. X X . . X X . X
X X . X . X . X X
. X X X X X X X .
(End)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Joshua Zucker, Mar 25 2009
EXTENSIONS
a(5)-a(7) from Robert Israel, Mar 25 2009
a(8)-a(9) from Heinrich Ludwig, Jul 01 2013
a(10) from Giovanni Resta, Jul 14 2013 (see A152125).
Reworded definition to align this with several similar sequences (A227133, A240443, A227116, etc.). - N. J. A. Sloane, Apr 19 2016
a(11)-a(13) (using A227133) from Alois P. Heinz, May 05 2023
STATUS
approved