# Fewest polyominoes adjacent to 3 copies

What is the smallest positive number of polyominoes P, such that

• You can place grid aligned copies of P without any overlap; and
• Each polyomino is adjacent to exactly 3 other polyominoes.

Polyominoes must be the same shape, but can be rotated or flipped. Two polyominoes are considered adjacent if they touch at one of their sides (not vertices).

• perhaps this is duplicate then? Jul 24, 2023 at 8:08
• It has the same answer, though it is a different question. Theoretically that makes it a duplicate, but I wouldn't insist on it. Jul 24, 2023 at 8:33

The minimum number of polyominoes needed is

4

Proof:

This nonomino is the smallest I could find that would work.

• Lovely construction. Four is the definitely the minimum because each polyomino needs three neighbours. Jul 24, 2023 at 2:22
• I wonder if we can find a smaller polyomino? Jul 24, 2023 at 3:14
• @DmitryKamenetsky There is an octomino here. Jul 24, 2023 at 6:46
• oh cool! That must be optimal Jul 24, 2023 at 6:59

Here is an example of 4 octominoes, which I believe is the smallest, and also the most compact:

Here Bass has another octomino, as pointed by @Jaap Scherphuis. The shape of his octomino is easier, however, it takes up more space than this one, according to the number of inner white squares.

• Very nice! Are there solutions without any inner space? What about smaller polyominoes? Jul 24, 2023 at 22:27
• I dont think it's possible to achieve this with smaller polyominoes or without any white space. Jul 25, 2023 at 4:42
• If these were physical pieces, would they hold together without falling apart? Jul 25, 2023 at 5:09
• Less white space is possible, see my answer Jul 25, 2023 at 5:20
• If you focus on just the whitespace, you're right. But the overall shape will be bigger, as well as each polyomino. I like your thinking :) Jul 25, 2023 at 7:01

I can reduce the number of inner spaces by 3, by adding 3 cells to each piece in @Lezzup's answer: