the two first moments of the number of clicks distribution.

Except in the limits $N_{\mbox{\tiny{y}}}=1$ (no yellow cells) and $N_{\mbox{\tiny{y}}}=N$ (yellow cells occupy the whole board), it is hard to obtain exact analytical expressions for the average total number of jumps needed to find the target. However, it is possible to obtain the distribution for the length of the extensive phase

$$\label{eq:extensive}
 P_i(N_{\mbox{\tiny{y}}}) = p_i(N_{\mbox{\tiny{y}}})\prod_{j=1}^{i-1}\left(1-p_j(N_{\mbox{\tiny{y}}})\right),$$

where $P_i(N_{\mbox{\tiny{y}}})$ is the probability of having an extensive phase of $i$ jumps when the neighborhood of the target has a lateral length $N_{\mbox{\tiny{y}}}$ and $p_i(N_{\mbox{\tiny{y}}})=\frac{N_{\mbox{\tiny{y}}}}{N-i+1}$ is the probability of finding a yellow cell in the $i-th$ mouse click. In words, the probability of having an extensive phase with $i$ jumps is given by the probability of not finding a yellow cell in all the previous movements multiplied by the probability of finding one in the $i-th$ movement. Given Eq. (\[eq:extensive\]), the mean length of the extensive phase is

$$\label{eq:limits-anal}
M_{\mbox{\tiny{ext}}}=\sum\limits_{i=1}^{N^{2}-N_{\mbox{\tiny{y}}}^{2}+1}iP_i(N_{\mbox{\tiny{y}}}).$$

For the length of the intensive phase however we can only give and upper and a lower limit, assuming that after the detection of the first yellow cell all the movements are to neighboring cells. Therefore, the target is found on average after $N_{\mbox{\tiny{y}}}^{2}/2$ 