Most replicators work accoding to a *parity rule:* there is
a (usually one-dimensional) grid of possible replicator positions,
such that, if replicators are placed at certain grid positions,
they produce new replicators at the positions with an odd number of
replicators in neighboring cells. If the grid positions are spaced
k units apart, and the period (number of generations between each
replication) is p, then the resulting pattern of oscillators
expands at a speed of kc/p, with a *sawtooth* growth rate in
which the number of active cells repeatedly increases to linear (or
quadratic, for two dimensional grids) then decreases to a constant.
The minima of the number of active cells occur at generations that
are powers of two times the period.

Parity-rule replicators are fairly common in rules with B1 (birth on a single neighbor, in standard semitotalistic notation), but those rules are not very interesting because any pattern quickly grows to infinity in all directions, so gliders, oscillators, and similar structures are impossible. The more interesting rules are those with B0, B2 or B3, in which gliders are possible (and many gliders are known). Among these rules, only the following replicators are known:

**B013456/S12356**4c/4 replicator**B013456/S2**3c/6 replicator**B0134567/S0124**5c/10 replicator**B013567/S01**7c/14 replicator**B01357/S012**3c/6 replicator breeder**B01357/S0123**2c/6 replicator**B013578/S0123**6c/20 replicator**B0136/S123**9c/30 replicator**B01367/S01**10c/20 replicator**B01367/S012**10c/20 and 11c/22 replicators**B01367/S0124**4c/8 two-dimensional replicator**B013678/S1**4c/8 replicator**B01368/S03**5c/10 replicator**B017/S12**4c/8 replicator**B01378/S123**9c/26 replicator**B0145/S01235**4c/8 replicator**B01468/S02**9c/20 replicator**B0147/S1**c/2 replicator**B015/S0**6c/12 replicator**B016/S12**6c/12 replicator**B016/S123**6c/16 replicator**B017/S016**3c/4 replicator**B017/S1**7c/14 replicator**B02/S4**5c/10 replicator**B024/S013**10c/20 replicator**B02345/S01235**4c/8 replicator**B02345/S023**5c/10 replicator**B023458/S023**9c/18 replicator**B02346/S023**(1,2)c/4 replicator**B02346/S13**4c/8 replicator**B02347/S0**3c/12 replicator**B0235/S0**multiple replicators**B0235/S23**4c/8 diagonal replicator**B02356/S34**6c/12 diagonal replicator**B0236/S0123**4c/10 replicator**B0236/S1**2c/4 diagonal replicator- B0236/S124 2c/4 diagonal replicator
**B0237/S0234**3c/8 replicator**B024/S013**10c/20 replicator**B0246/S1234**4c/10 replicator**B025/S05**4c/4 two-dimensional replicator**B026/S12**2c/4 diagonal replicator**B0267/S3**3c/6 replicator**B04/S2**c/2 diagonal replicator**B2/S13**3c/3 replicator**B25/S4**3c/3 replicator**B34/S0235**2c/4 replicator**B3568/S2347**2c/4 replicator**B357/S3458**5c/15 replicator**B3578/S23**4c/12 two-dimensional replicator**B358/S23467**4c/11 replicator**B36/S124**c/2 replicator**B36/S23**(HighLife) 2c/12 diagonal replicator**B36/S245**replicator**B368/S12578**4c/13 replicator**B37/S345**6c/35 replicator

It is believed that **B3/S23**
(Conway's Life) and **B35/S236**
also support replicators due to construction universality, but
explicit replicators for those rules have not been constructed.