Let us call a figure on a sheet of squares an arbitrary finite set of connected squares, i.e. a set of squares in which it is possible to go from any square to any other by walking only on the squares of this figure. Prove that for every natural n there exists such a figure on the sheet of squares that it can be cut into "corners" (Fig. 1) exactly in $F_n$ ways, where $F_n$ s the $n$-th Fibonacci number (in the series of Fibonacci numbers $F_1 = F_2 = 1$ and for each $i > 1$ holds $F_{i+2} = F_{i+1} + F_i$). For example, a rectangle of $2 \times 3$ squares can be cut at the corners in exactly two ways (Fig. $2$).