You don't. You'd find contemporaneous accounts from people who'd taken the test and complained about it. That's how the social science of history works. I think the consensus here is that the brain-teaser test is either not real, or was not widely used (nobody has been able to find an instance where it was).
A clarifying bit of context: there were extensive complaints about the multiple-choice constitutional interpretation tests that were given at the time.
> How hidden can you keep a test that many thousands of voters take?
After its taken, and presuming active measures were taken to prevent distribution other than for people taking it who would then return it, pretty easily. Paper is biodegradable, burns easily, can be shredded (and recycled into new paper), etc.
If there is a credible threat of retaliation (violence, employment, housing) for even trying to vote, then this is very effective. Why take a big risk if you won’t get to vote anyway? This way you don’t actually have to give the test very often, everyone quickly figures out the “rules” and falls into line.
