## Task 3 - 040733

Hans, Jürgen, Paul, and Wolfgang have taken the first four places in a 100-meter race. When asked who took the first, second, third, and fourth place, we received the following answers:

1. Paul first, Jürgen second;
2. Paul second, Wolfgang third;
3. Hans second, Wolfgang fourth.

In each of the three answers, one statement was true and one was false.

Who took the first, second, third, and fourth place?
First, consider statements 1 and 2. There are now 3 possibilities. Either Paul is first, or Paul is second, or Paul is neither first nor second.

Possibility 1: Paul is neither first nor second.

Then it would follow that Jürgen is second and Wolfgang is third. However, none of the statements in 3. can be true, because Hans cannot be second, as that is already Jürgen, and Wolfgang cannot be fourth, as he is already third, so this is not possible.

Possibility 2: Paul is second.

If Paul is second, then the statement "Paul is first" is false, so the statement "Jürgen is second" must be true. However, this is not possible because Paul is already second, so this is not possible.

Possibility 3: Paul is first.

So the statement "Paul is second" is false. It follows that Wolfgang is third (Statement 2). Therefore, the statement "Wolfgang is fourth" is false, so Hans is second (Statement 3). Jürgen is left and is thus fourth.

Paul took the first place, Hans the second, Wolfgang the third, and Jürgen the fourth. Given the solution path shown, there can be no other solution.