> Remember that, in this case, all I want to know is if they have that capability or not. Also, I don't care whether they can tell me how multiplication works or not, only if they can multiply two arbitrary numbers.
Right, but unless you know something about their process, the only way you can determine whether they have the ability to (correctly) multiply two arbitrary four-digit numbers is to have them demonstrate multiplying every combination of four-digit numbers. One can easily imagine a system that gets most answers correct but fails certain cases (e.g. only carrying between pairs of digits).
The reason why an inexhaustive test (using only several dozen examples) works to some degree with schoolchildren is because we know something about their method: the algorithm for multiplication they've been exposed to – that they're explicitly being taught – is an algorithm that we know to be correct.
That has a very large effect on the likelihood of different failure modes and the evidence required to have a particular degree of confidence.
Furthermore, for "system 2" activities (to borrow the term from Kahneman), we can reasonably expect a person's description of their process to match the process they actually performed. (There are exceptions: people will produced incorrect post hoc explanations for their behavior under sufficient duress when "I don't know" isn't perceived as an acceptable answer.) But I'm personally not aware of any reason to believe this about LLMs. I don't know why the network's actual process for performing multiplication should have anything to do with the text it produces after the fact when asked to explain its work.
So I'm not debating that a reasonable way to guess whether a person can (correctly) multiply two numbers is to "ask them to multiply two numbers and see if they get it right"; I'm disagreeing that this works with LLMs.