With the Thue-Morse sequence, people aren't simply taking turns. Rather, they're taking turns taking turns. The sequence distributes "advantage" more equally among the two parties than an alternating sequence, which ensures that the person who picks first has an advantage. It also has several interesting properties unrelated to turn-taking, such as its fractal nature.
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Key Facts In This Video
Here's how to use the Thue-Morse sequence to take turns: 00:32
If two people alternate when picking objects, then the person who goes first will always have the advantage. 02:08
The Thue-Morse sequence was independently discovered by a chess Grandmaster who figured out how to play an infinite game of chess. 06:10