Question 93
Let $k \geqslant 1$ be an integer.
Consider $4k$ tokens, $2k$ of which are red and $2k$ blue. A sequence of these $4k$ tokens can be transformed into another sequence by a move, consisting of interchanging some number (possibly just one) of consecutive red tokens with an equal number of consecutive blue tokens. For example, one move allows us to go from rbbbrrrb to rrrbrbbb, where $r$ denotes a red token and $b$ denotes a blue token.
Determine the smallest number $n$ (as a function of $k$) such that, starting from any initial sequence of the $4k$ tokens, at most $n$ moves are needed to reach the state in which the first $2k$ tokens are red.
Consider $4k$ tokens, $2k$ of which are red and $2k$ blue. A sequence of these $4k$ tokens can be transformed into another sequence by a move, consisting of interchanging some number (possibly just one) of consecutive red tokens with an equal number of consecutive blue tokens. For example, one move allows us to go from rbbbrrrb to rrrbrbbb, where $r$ denotes a red token and $b$ denotes a blue token.
Determine the smallest number $n$ (as a function of $k$) such that, starting from any initial sequence of the $4k$ tokens, at most $n$ moves are needed to reach the state in which the first $2k$ tokens are red.