I think that statement is still subject to the proof given. "Not all the dots are blue" just initialize K to {n}, and the induction proceeds as written. The fl

I think that statement is still subject to the proof given. "Not all the dots are blue" just initialize K to {n}, and the induction proceeds as written. The flaw in the proof is that it assumes that townsfolk only learn their dot color based on counting arguments. If the stranger's "anything non-trivial" statement includes information beyond just a raw count, then a townsfolk may learn their dot-color prematurely (e.g. by being told in the statement) and the proof fails.