> In other words do all NP problems contain structure that can be exploited to short-circuit the calculation or not. And we can strengthen this statement a bit

> In other words do all NP problems contain structure that can be exploited to short-circuit the calculation or not. And we can strengthen this statement a bit, to really hammer home the intuition of why P=NP would be such a radical discovery. "Simulating an arbitrary non-deterministic Turing machine for K steps" is itself an NP-complete problem, which would be solvable by a polynomial algorithm if P=NP. In other words, it's not just that all NP problems would have some structure that a sufficiently clever programmer could exploit; it's that there would exist a single algorithm that could automatically discover and exploit such structure in any NP problem.