A Simultaneous Search Problem 1

We introduce a new search problem motivated by computational metrology. The problem is as follows: we would like to locate two unknown numbers x, y ∈ [0, 1] with as little uncertainty as possible, using some given number k of probes. Each probe is specified by a real number r ∈ [0, 1]. After a probe at r , we are told whether x ≤ r or x ≥ r , and whether y ≤ r or y ≥ r . We derive the optimal strategy and prove that the asymptotic behavior of the total uncertainty after k probes is 13 7 2 −(k+1)/2 for odd k and 13 10 2 −k/2 for even k.