This study explores an online filling game. Given an initially empty array of length $n$, at each time step, an independent sample from $\text{Unif}[0,1]$ is observed, and the player must choose to either discard the sample or place it irrevocably into an empty position of the array while ensuring that the filled entries are non-decreasing from left to right.
We focus on the optimal expected time required to complete the array filling among all possible strategies, denoted as $v_n$. Our main result determines $v_n$ up to lower-order terms:
$$ v_n=\left(\frac{1}{2}+o(1)\right)n\text{log} n. $$
More precisely, no strategy, whether randomized or adaptive, can achieve an expected completion time lower than $\left(\frac{1}{2}-o(1)\right)n\text{log} n$. We provide an explicit deterministic strategy whose expected completion time is at most $\left(\frac{1}{2}+o(1)\right)n\text{log} n$.
In comparison, the natural coupon-collector strategy, which partitions $[0,1]$ into $n$ equal intervals and reserves one array position for each interval, has an expected completion time of $(1+o(1))n\text{log} n$.
Additionally, we consider a with-replacement version of the game, where previously placed entries may be overwritten. For this variant, we present a deterministic strategy with an expected completion time of $O(n\text{sqrt}(\text{log} n))$, thus establishing a separation between the two models.