number of clicks needed to find the target as the [*optimal interaction range*]{}. The standard deviation of the number of jumps is also minimal at the optimal range, which means a narrowing in the distribution of clicks used to detect the target and therefore a reduction in the stochasticity of the search. In the limit of zero information (i.e. no yellow cells or $Ny=1$, or the whole board is yellow, $Ny=N$), the probability of finding the target on the first click is given by the inverse of the number of available cells, $1/N^{2}$. In any subsequent movement, $m$, this probability is given by

$$\label{eq:limitnoinfo}
 P_m = \frac{N^{2}-(m-1)}{N^{2}}\times\frac{1}{N^{2}-(m-1)},$$

where the first term yields the probability of not having found the target in the previous $m-1$ clicks and the second term yields the probability of hitting the target once $m-1$ squares have been visited. Equation (\[eq:limitnoinfo\]) reduces to $1/N^{2}$ regardless of the value of $m$. Therefore, the probability of detection in the limit $N_{\mbox{\tiny{y}}}=1$ (and $N_{\mbox{\tiny{y}}}=N$) follows a uniform distribution of mean $N^{2}/2$ and standard deviation $N^{2}/\sqrt{12}$, which is in good agreement with data (black squares in Fig. \[fig:blindjumps\]).

Next, we analyze all the trajectories traced by the players in every round. 