  Consider a two-person zero-sum search game between a Hider and a Searcher.
The Hider chooses to hide in one of $n$ discrete locations (or "boxes") and the
Searcher chooses a search sequence specifying which order to look in these
boxes until finding the Hider. A search at box $i$ takes $t_i$ time units and
finds the Hider - if hidden there - independently with probability $q_i$, for
$i=1,\ldots,n$. The Searcher wants to minimize the expected total time needed
to find the Hider, while the Hider wants to maximize it. It is shown in the
literature that the Searcher has an optimal search strategy that mixes up to
$n$ distinct search sequences with appropriate probabilities. This paper
investigates the existence of optimal pure strategies for the Searcher - a
single deterministic search sequence that achieves the optimal expected total
search time regardless of where the Hider hides. We identify several cases in
which the Searcher has an optimal pure strategy, and several cases in which
such optimal pure strategy does not exist. An optimal pure search strategy has
significant practical value because the Searcher does not need to randomize
their actions and will avoid second guessing themselves if the chosen search
sequence from an optimal mixed strategy does not turn out well.
