You don't have to both shuffle the envelopes and flip a fair coin, but you have to do at least one of these or something equivalent. The first player chooses tw

You don't have to both shuffle the envelopes and flip a fair coin, but you have to do at least one of these or something equivalent. The first player chooses two numbers by any means whatsoever, but must pick one of them genuinely randomly. If the first player simply chooses two numbers and then tells you one of them arbitrarily, he/she could always pick the higher number -- or the lower number -- and you can't defeat both of these strategies independently of knowing which one he/she picked. BTW, I cheated and looked at the (official XKCD) solution -- and I recommend no one else do the same! This puzzle is awesome.