unsupported

I stand corrected. Thank you for pointing out my misunderstanding of US elections.

So what is the main difference? size of districts (constituencies) by a factor of ~10? Gerrymandering?

I think the reason why US has two parties is that most States apply FPTP at a district level, but then also have a winner-takes-all at State level based on the districts result. So one district selects one party, and then the State gives all its N votes to the majority party among districts, selecting N people of the same party. UK only has FPTP at constituency (district) level. So one constituency selects one person (one MP belonging to one party).

In UK when voting you have to chose among the top two parties in your own constituency in order to have a chance to get the MP you want rather than throwing your vote away. Votes distribution at national level doesn't really count from the voter point of view.

In US when voting you have to chose among the top two parties in your own State because if your district picks a third party and no other district does, your whole district vote is thrown away. Votes distribution at State level really counts from the voter point of view.

Reservoir sampling or Algorithm R. It allows you to randomly choose a sample of k items (with equal probability) from an "infinite" stream of items. At first it sounds impossible, but the solution and proof are extremely simple and elegant.