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It might be handy to memorize the multiplication tables of alternate bases. Some interesting ones might be (copied from [notes-abstract-categorizations-catChInts]):
(SHCN/CAN means Superior highly composite number/Colossally abundant number; the first 15 numbers in both of these sequences are the same. SHCNs are a subset of highly composite numbers. CANs are a subset of superabundant numbers. See also primorial. )
imo the multiplication table of anything much bigger than 10 will be too hard to memorize. So kill everything above 12 (and exclude 10 b/c we already know it):
how much effort would it take to learn these multiplication tables? The multiplication table has n^2 entries; but when one factor is 1 or n these are easy, so there's only (n-2)^2 entries to learn; and multiplication is commutative, so we only need the diagonal plus the elements below the diagonal; to get that from a square matrix with n^2 elements, if the number of elements is even (which occurs when n is even), we divide by two and add half of n; if n is odd, (i think, todo doublecheck) we subtract one and then do the same and then add back one. n-2 is odd iff n is odd. So the formula is ((n-2)^2 + n-2)/2 (i think). So the number of entries to memorize is:
so the total number to learn, without 12, would be: 0+1+3+6+10+15 = 35, which is about the same amount we had to learn to memorize the multiplication table for base 10. If we add in base 12, though, then that's 55 more to learn, bringing the total to 90, which is almost triple the effort we had to put in to learn the base 10 table. This is probably overstating the relative difficulty of base 12, however, probaby because there are probably a lesser proportion of numbers <12^2 which are coprime to 12, than numbers less than 100 that are coprime to 10 (todo explore this).
The logic behind these choices can be summarized as: 2 and 3 are unavoidably crucial; 6 is 3! and a primorial and SHCN and CAN; 5 is a prime smaller than 6; 4 is the surrounded by things we've got and it's cheap and it's superabundant and highly composite, so get it; now we have everything under 7 and memorizing 15 more entries is a reasonable price for the next prime and to have everything <=7; 12 is SHCN and CAN and Chernoff.
These tables (and more) can all be found on the first page of: http://www.dozenal.org/articles/DSA-Mult.pdf . On page 2 of that document you can see the "arqam" numerals for 10 and 11 (needed for base 12), which imo are better than Dwiggins.
(todo: explore other reasons why 12 might have lots of patterns; explore other numeral choices for 10 and 11)
