Come to think of it, this suggests that you might be able to save vigesimal by a sort of numeric diglossia with decimal. We could consider adding with the "teens" digits (that go down to the feet) to be an exercise that is covered only
after adding with single digits (on the hands) is completely understood, so that the gradation of addition facts would be:
1. The trivial facts (adding 0 and 1);
2. The facts whose answer doesn't surpass a;
3. The facts whose addends don't surpass a;
4. The facts where one addend surpasses a;
5. The facts where both addends surpass a.
The beginning, in fact, would look exactly like decimal, except that once we get to group 3, we're writing "8 + 6 = e" instead of "8 + 6 = 14". Of course we'd still
say "fourteen" or something like that, but we'd write it with its own transdecimal digit. Starting group 4 would presumably be facts like "plus a, without carries", "plus 5, without carries", and "plus f, without carries", as
Oschkar's exposé has them. (I may write a revised version going through this route.) Group 5 can either round off from 10 using subtractive strategies or simply group the "teens" together into a carry and then use group 3 strategies for the remainders.
In this sort of vigesimal, you'd learn the tables up to ten times ten, but in base twenty, and you could have a lot of code-switching involved with a decimal nomenclature. So you could say "eight nines make seventy-two", but you'd always write it as "8*9=3c". Once again we have:
1. The trivial facts (multiplying by 0 and 1);
2. The facts whose multiplicands don't surpass a;
3. The facts where one multiplicand surpasses a;
4. The facts where both multiplicands surpass a.
I imagine that the effect of multiplying by a (halving a number) has to be taught early, somewhat apart from these groups. Then we can learn the tables up to a*a=50 using something close to the decimal strategies (although I confess I haven't really thought much about the alterations needed because the base is double the size). To do the facts in group 3, you break up the larger multiplicand into a and some leftovers; so g*7 = a*7 + 6*7 = 3a + 22 = 5c. (I think the facts with a need to be thoroughly drilled in, so that they feel almost as round as the true scores. Perhaps there needs to be finger-only counting as well as finger-and-toe counting?) To do the facts in group 4, you use the quarter-squares identity, just as you can for the high facts in group 2 if the decimal nine-times strategies aren't obvious enough. For example, d*e = (a+3)(a+4) = (h*a) + (3*4) = 92. Of course you can in some cases use 10 instead, like in g*h = (10-4)(10-3) = d0 + c = dc. And if you do them enough, it might well be possible to memorise them all.
I'm not sure if this can really be taken past twenty, because of the need to learn lots of different numerals. But if it can, it seems to be one of the best hopes available for such a double-sized base.
P.S. I think I've forgotten the even bigger problem that this doesn't really reduce the number of facts you have to learn, but only it makes it possible to learn the ones that are already there, so that I think it's massively unlikely that this would actually work to get people to memorise the facts for {24, 28}. This being said, these bases are larger, so it takes less efficiency to get them to work as long as the bottom of the range is well-understood. I think I'll have to try them all out again. This is honestly still using mid-scale strategies with the exception that removing the alternation perhaps removes the biggest hurdle, but they might not feel that way.