jumps in the intensive phase when the neighborhood of the target is large and after $(N_{\mbox{\tiny{y}}}+2)^{2}/2$ movements when the neighborhood is small. These two limits account for the decreasing probability of visiting cells outside the neighborhood when increasing its size. For small values of $N_{\mbox{\tiny{y}}}$ it is very likely to reach the border of the neighborhood before detecting the target and thus to return to the black region. Combining these two results for the intensive phase with the length of the extensive phase obtained in Eq. \[eq:limits-anal\], we obtain two theoretical approximations to the total number of clicks

$$\begin{aligned}
 M^{\mbox{\tiny{up}}} &=& \sum\limits_{i=1}^{N^{2}-N_{\mbox{\tiny{y}}}^{2}+1}iP_i + \frac{(N_{\mbox{\tiny{y}}}+2)^{2}}{2}, \\
 M^{\mbox{\tiny{low}}} &=& \sum\limits_{i=1}^{N^{2}-N_{\mbox{\tiny{y}}}^{2}+1}iP_i + \frac{N_{\mbox{\tiny{y}}}^{2}}{2}.\end{aligned}$$

The combination of these two expressions gives an approximated range for the length of the search (magenta region in Fig. \[fig:blindjumps\]) that shows an excellent agreement with empirical data and numerical simulations of the model.

Model for searches with initial information. The design of optimal strategies. {#model-for-searches-with-initial-information.-the-design-of-optimal-strategies. .unnumbered}
------------------------------------------------------------------------------

Knowing the size of the yellow region at the beginning of the game changes the nature of the search as the information gathered by the player with each movement may be used to design the next displacement. This reinforces the non-Markovian 