In the classic house robber problem, you are given an array of numbers and you have to choose a subset with maximum sum subject to not picking adjacent numbers. The classic solution uses DP.

Consider now a version where, for each number, you can pick a fraction of it, ...

but the fraction you take from two consecutive numbers cannot exceed 100%. So, if you take 60% from a number, you can only take 40% from adjacent ones.

Is there an intuitive explanation for why there's always an optimal solution that only uses 100% or 0% for each number?