Read the $2n+1$ colours from left to right. If every neighbouring pair had the same colour, <mathematical induction> would make every point the same colour as $-1$, contradicting the different colour of $1$. Therefore at least one adjacent pair has different colours.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 11H / a / Solution

 Solution (source code)