I have proven that the family of coiled snakes and the block are the only width-2 strict SLs in Life.
PROOF
Let us denote the following columns of cells as so:
These are the columns of a width-2 SL.
By convention, I do not allow 0 at the beginning. Hence the beginning is either 1, 2, or 3.
However, 1 or 2 cannot be the left terminus of a width-2 SL. Proof without words:
Code: Select all
x = 20, y = 8, rule = LifeHistory
AB4.2A4.AB4.2A$2B4.2B4.BA4.BA5$DB4.DA4.DB4.2A$2B4.2B4.BA4.CA!
Reflect this diagram to give the corresponding for nonexistence of left terminus 2.
Hence column 3 must be the beginning of a width-2 SL.
The column sequence 30 is not allowed as it gives a domino, which dies.
Sequence 33 gives the block, and since the rightmost cells have S3, the only allowed next column is 0.
Sequences 31 and 32 are the same up to reflection, so I will only consider 31.
31 produces a B3a, but this can be fixed by making the next column a 2. Making it a 0 will not work because of B3a, 1 because of B3i, and 3 because of no S4i. Hence 2 is the only option that would give a SL.
0 is not allowed as the next column because of no S1c, 1 because of B3e, 2 because of B3j. Hence 3 is the only viable option. Of course, one can terminate it here, giving the snake.
Can we continue this? Of course. Based on the reasoning above, the next viable columns are 1, 2, 3, 1, 2, 3... and so on ad infinitum. One can terminate the infinite SL at the columns '3' to give the coiled snakes.
Hence block and the coiled snakes are the only width-2 strict SLs. QED.
EDIT: request by confocaloid