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Nim multiplication is cool, but what does it have to do with Galois?
Apparently finite fields are also called Galois fields :)
Fair. =) If you're interested in more crazy game theory, I will (probably for the millionth time) recommend Conway's "On numbers and games", it's a very interesting and entertaining read.
Nim multiplication is simply not humanly possible to solve during contest. But it was posed on New Year Prime Contest 2017/2018, so...
Interestingly, this problem was in Vietnamese OJ since 10 years ago, and several solved it including some high school students =).
There is also English version of it