What is Invariant method?

Example 12.3 (The hat line)

With $k$ colours numbered $0, \dots, k-1$, the last person in the line (who sees everyone else) announces the sum of the colours in front modulo $k$. Each following person knows that sum, hears every answer given after the first, sees everyone in front, and so can subtract to find their own colour. All but the first are right, whatever $k$: the invariant is the announced sum.