A fun math puzzle #mathsky:
In front of you is a line of blue and green tokens. At each turn, you choose an adjacent pair of a blue token followed by a green token; your opponent replaces them with any number of green tokens followed by any number of blue tokens.
In front of you is a line of blue and green tokens. At each turn, you choose an adjacent pair of a blue token followed by a green token; your opponent replaces them with any number of green tokens followed by any number of blue tokens.
Comments
s=GᵐBⁿ
that replaces BG fixed (i.e. same every turn)?
If yes, I think a necessary and sufficient condition is that either
min(m,n)∈{0,1}
or the initial sequence is something like
(GG...GG)BG(BB...BB).
min(sup(m),sup(n))∈{0,1}
instead, where the supremum is taken over the set of all possible sequences s allowed.
(At any rate, @joeldavidhamkins.bsky.social might like this game.)