Children at the camp threw a die and, depending on the result, performed the following tasks:

| 1 | go $1 \mathrm{~km}$ west |
| :--- | :--- |
| 2 | go $1 \mathrm{~km}$ east |
| 3 | go $1 \mathrm{~km}$ north |
| 4 | go $1 \mathrm{~km}$ south |
| 5 | stand still |
| 6 | go $3 \mathrm{~km}$ north |

After five throws, Mark's team was $1 \mathrm{~km}$ east of the starting point.

1. What route could Mark's team have taken? Indicate at least four possibilities.
2. What could the total sum of all the numbers that fell for this team be? Determine all possibilities.

(E. Semerádová)

Hint. Which combinations of throws will cancel each other out in their consequences?
Since Mark's team ended up $1 \mathrm{~km}$ east of the starting point, they must have rolled a two at least once. The other four rolls must have been such that after completing the corresponding tasks, the team was back at the original location (e.g., $3 \mathrm{~km}$ north and $3 \mathrm{~km}$ south). All such quartets of rolled numbers, disregarding the order, are as follows:

$$
1122, \quad 1234, \quad 1255, \quad 3344, \quad 3455, \quad 4446, \quad 5555 .
$$

Together with the above-mentioned two, we get all possible sums of the numbers that could have been rolled by this team:

$$
8, \quad 12, \quad 15, \quad 16, \quad 19, \quad 20, \quad 22 .
$$

The number of possible routes that Mark's team could have taken is countless. For example, the sequence of rolls 24446 corresponds to a route of $1 \mathrm{~km}$ east, three times $1 \mathrm{~km}$ south, and $3 \mathrm{~km}$ north. By changing the order of these rolls, 19 more possibilities can be obtained, making a total of 20 different routes. In other cases, it is similar, except that different orders may represent only different ways of traversing the same route (see, for example, the orders $12122, 21122, 12212$). If a five appears in the sequence of rolls (staying in place), the number of possible routes significantly decreases.