I was trying to work out the smallest majority required to pass a Constitutional amendment. I expected the answer to be 2/3rd or 3/4ths, but came up with 11%:
NOTE: in the below, I am referring to state legislators, not US Senators or US Representatives.
It requires "the Legislatures of three fourths of the several States" to pass (ratify) an amendment. I realize no amendment has actually been passed this way, but it is theoretically possible.
There are 50 states. Three fourths of 50 is 37.5, so we round up to 38 states.
The 38 least-populated states make up 42% of the entire US population. Like in the Senate (and unlike in the House), size does not matter. Wyoming counts the same as California.
Most (all?) of these 38 states are broken up into legislative districts. You need to win only 51% of these districts to pass the amendment. Legislative approval is by simple majority, and the governor does not participate.
In each of these 51% of all districts, you need only 51% of the vote to elect a legislator.
The 51% least populated districts in a state make up at most 51% of the population (probably less, since legislative districts have approximately the same number of people, but not exactly).
Therefore, you need at most 51% of the people in 51% of the districts for a state to pass an amendment. That's about 26% of the people in a state.
Therefore, you need 26% of the people in 38 states (making up 42% of the population) to pass an amendment. That works out to about 11%
What, if anything is wrong with this calculation? Several notes/disclaimers:
I'm using "51%" above to mean "more than 50%". In districts with a million people, even one vote over 50% works.
I realize this can't be done realistically, so this is primarily a theoretical question.
11% is close to an absolute extreme, but what if 30% of the population supported an amendment, and at least some of those people were willing to move or declare residency in other areas, and the entire effort was coordinated through the Internet. Would this have a real chance of working? I strongly suspect the 70% opposed would amend Article V itself, but it would be an interesting project.
Not all states have 100+ member legislative districts (although most do), so "51%" in this case means "more than half".
EDIT: A joke answer to this question: you need roughly 2000 people, provided they are all state legislators or something similar. Obviously, that's not the answer I'm looking for.
EDIT: A "realistic" way this could happen (to answer @NL7 concerns):
Suppose 11% of the US electorate is zealously determine to pass an amendment, and are willing to put a reasonable amount of time, effort, and money into doing so.
The 11% rearrange themselves (moving and/or declaring residency) to form the majorities I describe above.
One of the 11% then runs for various state legislator offices. This is one of the zealots running for office: he doesn't have a party affiliation and his greatest ambition is to pass this amendment, not to achieve higher office.
By the argument above, the amendment would pass.
In other words, the zealous 11% chooses the candidate. They don't just hope that the existing candidate follows the majority vote.
Keep in mind Constitutional amendments can significantly change political power: the 11%ers might end up giving themselves a lot more political power than they would have than just running for higher office.
NB: The word "gerrymandering" does not appear in this question.