Ken's Puzzle of the Week

Unique 3x3 Grids

               
               
               
               
               
               
               
               

   In an 8x8 grid of squares, blacken as few unit squares as possible to make each 3x3 grid of nine squares unique (There are 36 3x3 grids.)  For example, in a 5x5 grid of squares, you need to make only one square black.

Repeat for 6x6 thru 10x10 grids of squares.

Source: Original.
Solutions were received from Tony Wang, Dan Chirica, Kirk Bresniker, and Keith Lynch.  Dan Chirica provided solutions for all, but I don't know if they are minimal.   9x9 has been shown to have an upper bound of 14 and lower bound of 12 (no solutions exist with only 11 squares filled.)  Keith's first solutions are below: 
For six on a side, the minimum is four.  there are 18 solutions, excluding rotations and reflections.
The most symmetrical is:

- - - - - -
- - - - - -
- - * * - -
- - * * - -
- - - - - -
- - - - - -

For seven on a side, the minimum is six.  There are 11 solutions, none of them symmetrical.  Here's one of them:

- - - * - - -
- - - - - - -
- - * - - - -
- * - - * - -
- - - - - * -
- - * - - - -
- - - - - - -

For eight on a side, the minimum is eight.  There are two solutions, one with symmetry through the diagonal,
one without.  They are very similar.

- - - - - - - -     - - - - - - - -
- - - - - - - -     - - - - - - - -
- - * * - * - -     - - * * - * - -
- - * - - * - -     - - * - - * - -
- - - - - - - -     - - - - - - - -
- - * * - * - -     - - * - * * - -
- - - - - - - -     - - - - - - - -
- - - - - - - -     - - - - - - - -
I notice that for five, six, eight, and nine on a side, there are solutions with the outer two rows and columns empty. For five and eight, *all* solutions have that property. But for seven on a side, *none* of the solutions are in this form.
Dan Chirica found a 9x9 solution with only 13 blacks and a 10x10 with only 20 blacks. I don't know if they're minimal, but they're the smallest so far. 
9x9, 13 blacks
 - - - - - - - - -
 - - - - - - - - -
 - - 1 1 1 - 1 - -
 - - 1 - - - 1 - -
 - - - 1 - - - - -
 - - 1 - 1 - - 1 -
 - - - - - 1 1 - -
 - 1 - - - - - - -
 - - - - - - - - -
10x10, 20 blacks 
  - - - - - - - - - -
  - - - - - - - - - 1
  - - - 1 - 1 1 - - -
  1 1 1 - - - - - - 1
  - - - - 1 1 - 1 - -
  - - 1 - - 1 - - - -
  - - 1 - - - - 1 - -
  - - 1 - 1 - 1 1 - -
  - - - - - 1 - - - -
  - - - - - - - - - -
Kirk Bresniker provided this complete list of solutions up to 8x8: 
5 with 1 mark: 1 solution
00000 00000 00100 00000 00000

6 with 4 marks: 18 solutions
010000 000000 000100 001100 000000 000000
010000 000000 000100 001000 001000 000000
010000 000000 000100 001000 000000 001000
010000 000000 000100 000100 001000 000000
001000 001000 000000 001100 000000 000000
001000 001000 000000 000100 001000 000000
001000 000000 001000 001100 000000 000000
001000 000000 001000 000100 010000 000000
001000 000000 001000 000100 000000 001000
001000 000000 000100 001000 001000 000000
001000 000000 000100 000100 010000 000000
001000 000000 000100 000100 001000 000000
000000 010000 000110 001000 000000 000000
000000 010000 000100 001100 000000 000000
000000 010000 000100 000100 001000 000000
000000 001000 001000 001100 000000 000000
000000 001000 001000 000100 001000 000000
000000 000000 001100 001100 000000 000000

7 with 6 marks: 11 solutions
0001000 0000000 0010000 0100100 0000010 0010000 0000000
0001000 0000000 0010000 0010100 0000010 0010000 0000000
0001000 0000000 0010000 0000100 0010010 0010000 0000000
0000000 0100000 0010100 0000100 0000000 0011000 0000000
0000000 0100000 0001100 0000100 0010000 0001000 0000000
0000000 0010000 0100100 0000100 0010000 0001000 0000000
0000000 0010000 0010100 0000010 0011000 0000000 0000000
0000000 0010000 0010100 0000010 0001000 0010000 0000000
0000000 0010000 0010010 0100000 0001100 0000000 0000000
0000000 0010000 0010010 0000100 0010000 0001000 0000000
0000000 0010000 0000100 0010010 0011000 0000000 0000000

8 with 8 marks: 2 solutions
00000000 00000000 00110100 00100100 00000000 00110100 00000000 00000000
00000000 00000000 00110100 00100100 00000000 00101100 00000000 00000000
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