# Ten tetrominoes inside an 8x8 grid

Can you place ten tetrominoes inside an 8x8 grid, such that they do not overlap or touch each other orthogonally (horizontally or vertically) ?

• Another way of articulating this is to find an 8 x 8 pixel binary image for which performing connected-component labelling results in exactly 10 components of area 4.
– Wyck
Commented Jun 13, 2022 at 15:38
• ok I'll wait then. But I can't give the tick to all 3... Commented Jun 14, 2022 at 11:03
• The solution already existed on the site puzzling.stackexchange.com/questions/90126/… Commented Mar 22 at 20:41

As promised, here are the three solutions that my computer found.

The first was already found by franck vivien (and Bass), the second by JLee.

what about the following arrangement of

8 T tetrominoes + 2 squares?

I found it manually by searching arrangement of same pieces, then had to change a bit strategy

• Well done! Can you find any other solutions? Commented Jun 13, 2022 at 7:46
• I have confirmed by computer that there are two other solutions (ignoring rotation/reflection). If no one finds these, I'll post them in a day or two. Commented Jun 13, 2022 at 10:32
• @JLee You are talking about the independent domination number. You might be interested in my answer to this closely related problem. Commented Jun 13, 2022 at 15:11
• @JLee To see that $4$ is a lower bound for the independent domination number, note that no two of the following four tetrominoes can be blocked by a single tetromino: $$\begin{matrix} 1 &. &. &. &. &2 &2 &2 \\ 1 &1 &. &. &. &. &2 &. \\ 1 &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &. \\ . &. &. &. &. &. &. &3 \\ . &4 &. &. &. &. &3 &3 \\ 4 &4 &4 &. &. &. &. &3 \\ \end{matrix}$$ Commented Jun 13, 2022 at 20:20
• – JLee
Commented Jun 13, 2022 at 20:29

Now that I realize we don't have to use only T's, here is one solution:

• Well done. This is the original solution I had in mind. Now there is one more solution left. Can you find it? Commented Jun 13, 2022 at 22:11
• I tried for about 3 hours. Gotta take a break. Thx for the fun puzzle.
– JLee
Commented Jun 13, 2022 at 22:23