In
Conway's life, any
stable pattern, usually assumed to be
finite
and nonempty. For the purposes of
enumerating still lifes this
definition is, however,
unsatisfactory because, for example, any
pair of
blocks would count as a still life, and there would therefore
be an
infinite number of 8-bit still lifes. For this reason a
stricter definition is often used, counting a stable pattern as a
single still life only if its
islands cannot be
divided into two
nonempty sets both of which are stable in their own right. Compare
pseudo still life.
The requirement that a still life not be decomposable into two
separate stable patterns may seem a bit arbitrary, as it does not
rule out the possibility that it might be decomposable into more
than two. This is shown by the patterns in the following diagram,
both found in September 1998. On the left is a 33-cell pattern
(by Matthew Cook) that can be broken down into three stable pieces
but not into two. On the right is a 44-cell pattern (by Gabriel
Nivasch) that can be broken down into four stable pieces but not
into two or three. (Note that, as a consequence of the Four-Colour
Theorem, four is as high as you need ever go.) It is arguable that
patterns like these ought not to be considered as single still lifes.
..................O......
..............OO.O.O.....
..OO.OO.......O.OO.OOO...
.O.O.OO...............O..
.O.............OO.OO.OO..
OO..OO.O.......O...O.O...
O...OO.OOO......O.O...O..
.O........O....OO.OO.OO..
..O......OO....O.........
...O.OO.O.......O.OO.OO..
....OO.OO......OO.O.OO..O
.......................OO
Still lifes have been enumerated by Conway (4-7 bits), Robert
Wainwright (8-10 bits), Dave Buckingham (11-13 bits), Peter Raynham
(14 bits) and Mark Niemiec (15-24 bits). The resulting figures are
shown below. (These figures shouldn't be affected by the above
discussion of the strict definition of "still life", because
it is unlikely that there are any doubtful cases with less than 33 cells.)
4 2
5 1
6 5
7 4
8 9
9 10
10 25
11 46
12 121
13 240
14 619
15 1353
16 3286
17 7773
18 19044
19 45759
20 112243
21 273188
22 672172
23 1646147
24 4051711
Source: Life Lexicon by Stephen Silver
url: http://www.cs.jhu.edu/~callahan/lexiconf.htm