What is the optimal Jewish toenail cutting algorithm?

You could generate all possible toenail cutting sequences with no restrictions, and then filter out all sequences that violate the jewish rule. Luckily, humans only have five toes per foot*, so there are only 5! = 120 unrestricted sequences.

Python example:

#seq is only valid when consecutive elements in the list differ by at least two.
def isValid(seq):
    for i in range(len(seq)-1):
        a = seq[i]
        b = seq[i+1]
        if abs(a-b) == 1:
            return False
    return True


from itertools import ifilter, permutations
validseqs = ifilter(isValid, permutations([1,2,3,4,5]))
for i in validseqs:
    print i

(1, 3, 5, 2, 4)
(1, 4, 2, 5, 3)
(2, 4, 1, 3, 5)
(2, 4, 1, 5, 3)
(2, 5, 3, 1, 4)
(3, 1, 4, 2, 5)
(3, 1, 5, 2, 4)
(3, 5, 1, 4, 2)
(3, 5, 2, 4, 1)
(4, 1, 3, 5, 2)
(4, 2, 5, 1, 3)
(4, 2, 5, 3, 1)
(5, 2, 4, 1, 3)
(5, 3, 1, 4, 2)

To enforce your “no repeats of the same sequence” rule, you can just choose four of the above sequences, and use them alternately. The only catch here is that if you count the two big toes as “consecutive”, then you can’t choose two sequences that end and begin with 1, respectively.

*You may want to make a numberOfToesPerFoot variable, so you can easily change it later if any of your clients turn out to have less toes than you expect, or more.