Help choosing a system for mental math

[Kinma]

there didn’t seem to be a consensus about which mental math system is best.

IMHO, there just isn’t any system that is best.
Depending on the numbers given there is a generic method and there are shortcuts.

I’m looking for a system that allows for rapid multiplication of large numbers (maybe up to 4 x 4) and computation of square roots (and perhaps other fractional powers), and I’d like to do this without looking at written numbers for reference. I’d also (ideally) like the number to be produced from left to right, if possible.

Start with the criss cross method. This is the most generic one.
Then learn the shortcuts. See my old posts for examples.

I’m leaning towards Arthur Benjamin’s method (although I’ve never seen him compute square roots), but my concern is that I’ll invest 100 hours or so in one method just to find out that it’s deficient in one particular area, or an entirely different method is superior.

Arthur Benjamins system is great.
Definitely worth learning an using.

Also learn ‘difference of squares’.

Imo, it is best to start out using all the systems and calculating a multiplication using them all.

So for example:
23 X 37

Arthur Benjamin:
20 X 40 + 3 X 17 = 800 + 51 = 851

Difference of squares:
30^2 - 7^2 = 900 - 49 = 851

Criss cross:
20 X 30 + 20 X 7 +30 X 3 + 3 X 7 = 600 + 140 + 90 + 21 = 851

Also learn to do the 9 proof and 11 proof .
This will tell you wether your answer is correct.

9 proof
23 mod 9 = 5
37 mod 9 = 1
5X1=5

851 mod 9 = 5

11 proof:
23 mod 11 = 1
37 mod 11 = 4
1X4 = 4

851 mod 11 = 4

If both the 9 proof and the 11 proof come out correct you can be certain that your answer is correct.

Also; it builds your mind to be able to juggle large numbers.
Something you need when you get to 20x20.

Let me know if you need more info.
I’ll be happy to explain any of the methods used above.