Question 8
A sequence $x_1,x_2,\ldots,x_n,\ldots$ consists of an initial block of $p$ distinct positive integers which then repeats periodically. Thus $x_1,x_2,\ldots,x_p$ are $p$ pairwise distinct positive integers, and $x_{n+p}=x_n$ for every positive integer $n$.
The terms of the sequence are unknown, and the goal is to determine its period $p$. At each step, you may reveal the value of any one term you choose; your choice may depend on the results of previous steps.
(a) Knowing in advance that $1\leq p\leq10$, determine the smallest $n$ for which there is a strategy that determines $p$ with certainty by revealing at most $n$ terms.
(b) Knowing in advance that $p$ is one of the first $k$ primes, determine for which values of $k$ there is a strategy that determines $p$ with certainty by revealing at most five terms.
The terms of the sequence are unknown, and the goal is to determine its period $p$. At each step, you may reveal the value of any one term you choose; your choice may depend on the results of previous steps.
(a) Knowing in advance that $1\leq p\leq10$, determine the smallest $n$ for which there is a strategy that determines $p$ with certainty by revealing at most $n$ terms.
(b) Knowing in advance that $p$ is one of the first $k$ primes, determine for which values of $k$ there is a strategy that determines $p$ with certainty by revealing at most five terms.