By far the most symmetric restriction of a rule table is to consider
only the number of neighbors of a cell, but not their spatial
distribution. The result is called a totalistic rule, because of
its dependence on the sum. For the 
Moore neighborhood, there are
nine neighbors, therefore ten different sums. Sums of 0 and 9 can occur
in only one way, but there are nine neighborhoods containing but one
cell, and nine more with eight neighbors. In general, the binomial
coefficients 
tell how many neighborhoods contain k
cells.
A menu display for defining a totalistic rule will typically display the various sums along a row, the value which the cell will acquire in the new generation being shown in the matching position in the row below. The position of a movable cursor shows where new values may be inserted, thereby changing the rule.