TRICKY PROBLEM: A couple of friends want to rent an apartment. The rooms are quite different and the friends have different preferences and different ideas about what's worth what. Is there a way to split the rent and assign rooms to the friends so that everybody ends up being happy? In this video the Mathologer sets out to explain a very elegant new solution to this and related hard fair division problems that even made it into the New York Times. Featuring Sperner's lemma and Viviviani's theorem. Check out 3Blue1Brown's video on another fair division problem here: https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw Francis Su's article in the American Mathematical Monthly on which this video is based lives here https://www.math.hmc.edu/~su/papers.dir/rent.pdf You can find his fair division page here https://www.math.hmc.edu/~su/fairdivision/ To find the New York Times article "To Divide the Rent, Start With a Triangle" just google this title (the url is ages long and I don't want to reproduce it here). The NY Times fair division calculator. https://www.nytimes.com/interactive/2014/science/rent-division-calculator.html A proof of Brouwer's fixed-point theorem using Sperner's lemma www.math.harvard.edu/~amathew/HMMT.pdf Enjoy :) P.S.: One more thing you can think about is the following: how can what I show in the video be used to prove Viviani’s theorem.