16. A flea moves by jumping back and forth along a straight line. The flea's den is a point on the line. The jumping rules are as follows:

- if the flea is at a distance less than or equal to one meter from the den, after the next jump, it will be at a distance double the previous one, moving further away from the den.
- if the flea is at a distance $d$ greater than one meter from the den, after the next jump, it will be at a distance $\frac{1}{d}$ from the den but on the opposite side from where it is currently.

If after 5 jumps the flea is $80 \mathrm{~cm}$ from the den in a certain direction, in how many distinct sequences of jumps could it have reached that position?
16. The answer is 6. First, observe that regardless of the initial position, after the first jump, the flea will be at a distance less than or equal to 2 meters from the burrow, and with subsequent jumps, it cannot reach positions more than two meters from the burrow.

Next, note that according to the jumping rules, a point at a distance \( x \) less than \( 1 / 2 \) meter or greater than or equal to one meter from the burrow can only be reached with a jump from the point at a distance \( x / 2 \) from the burrow.

Positions that are \( 1 / 2 \leq |x| < 1 \) from the burrow can instead be reached either from \( x / 2 \) or from \( -1 / x \). Tracing the flea's path backward, these will be positions from which 2 possible backward trajectories can start.

Let's determine the positions the flea could have been in after the first jump: the final position of the flea is at \( 4 / 5 \) meters from the burrow. This is a branching point: the final position can be reached from either \( 2 / 5 \) meters or from \( -5 / 4 \) meters.

In the first case, \( 2 / 5 < 1 / 2 \) and can only be reached with the sequence \( 1 / 20, 1 / 10, 1 / 5, 2 / 5 \). At \( -5 / 4 \), it can only be reached from \( -5 / 8 \), which is a branching point: it can be reached from \( -5 / 32, -5 / 16 \) or from \( 4 / 5, 8 / 5 \).

In summary, there are three possible positions after the first jump (and thus 3 possible sequences of jumps) that allow reaching \( 4 / 5 \):

\[
\frac{4}{5}, \frac{8}{5}, -\frac{5}{8}, -\frac{5}{4}, \frac{4}{5} ; \quad \frac{1}{20}, \frac{1}{10}, \frac{1}{5}, \frac{2}{5}, \frac{4}{5} ; \quad -\frac{5}{32}, -\frac{5}{16}, -\frac{5}{8}, -\frac{5}{4}, \frac{4}{5}.
\]

The initial position of the flea is not subject to any constraints, in particular, it is not necessarily a position that can be reached with a jump. This means that each of the three positions after one jump can be reached from initial situations, that is, respectively, coming from

\[
\frac{2}{5} \text{ and } -\frac{5}{4} ; \quad \frac{1}{40} \text{ and } -20 ; \quad -\frac{5}{64} \text{ and } \frac{32}{5}
\]

The possible sequences are thus 6. What is described is summarized in the figure on the following page.

![](https://cdn.mathpix.com/cropped/2024_04_17_683cb69a22261aabc287g-12.jpg?height=753&width=997&top_left_y=146&top_left_x=538)