The number sequence $1, 9, 8, 2...$ satisfies the following rule: each element of the sequence starting from the fifth, is equal to the last digit of the sum of the previous four members. Will we ever meet four successive members equal to $3, 0, 4, 4$.
My intuition said that this was a invariance problem. So I searched for the invariant. The only obvious invariant I could come up with was remainders modulo 2. These remainders form a sequence which repeats: $1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1...$ This invariant (maybe more of a semi-invariant) does not give a different result for the required sequence. I could not devise another invariant to solve the problem.
I tried other remainders: modulo $3$, modulo $5$, etc, but nothing yields the answer.