The odds are indeed 1 in 243 that the players will "tie" 5 straight times in a 5-throw game of rock, paper, scissors.
The posters who were coming up with 243 x 243 are counting the total number of different ways that 5 rounds of rock, paper, scissors can be thrown in a two player game. But only 243 of those ways are 5 straight "ties"; in all the rest of the possible different 5-throw games, the players don't throw the same thing on at least one of the five throws.
Now, as extra credit, and to test for understanding of probabilities: What are the odds that in a five-throw game of rock, paper, scissors, the players will NOT choose the same thing on any of the 5 throws (i.e. there will be no "ties")?
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