Let $K_n$ be an $n$-tuple with elements as the first n non negative numbers. For example $K_5$ can be $(2,3,1,4,0)$.
Let a "shift" be a rotation in the tuple, i.e. after a "shift" the $K_5$ from before becomes $(0,2,3,1,4)$.
If we have $K_n$ for the first m primes, i.e. $K_2,K_3, ...$ m terms. What is the maximum number of shifts we would need to say for sure that at some point the first term of all the tuples was not $0$.
EDIT: When shifting, all tuples shift simultaneously