First, you cut the rope at 25 meters and make a loop at the end of the short part. Then you pass the longer part through the loop so that you get a 50-meter long rope (the second 25 meters are doubled). You use this rope to descend to the pin and grab it. Then tie one of the ends of the 50-meter rope to the pin, pull the rest back through the loop and let it fall to the ground.

The solution is shown below.

8 moves example:
1. f3 Nf6
2. Kf2 Ng8
3. Ke3 Nf6
4. Kd3 Ng8
5. Ke4 Nf6
6. Ke3 NNg8
7. Kf2 Nf6
8. Ke1 Ng8
In order to see that 8 is the minimum number of moves, notice that Black could only move rooks and knights, and therefore he has made an even number of moves. This implies that White has made an odd number of moves, excluding the pawn on f3. This is possible only if he has placed his king on a white cell at some point and then returned it back to e1, which would take at least 8 moves.

The cracks of the left gunshot end up right at the cracks of the right gunshot. Therefore the first gunshot is the one on the right.

The solution is shown below.

The reason is that as being taller, John also had longer arms. If Jack climbed on John’s shoulders, he wouldn’t have reached the key.

e3 Rxe3
c3 Rxc3
Ka2 Ra3+
Kb1 Ra1+
Kc2 Rc1+
Kd3 Rc3+
Ke2 Re3+
Kf1 Re1+
Kg2 Rg1+
Kf3 Rxg3+
Ke2 Re3+
Kd1 Re1+
Kc2 Rc1+
Kb3 Rc3+
Ka2 Rxc7
Rh8+ Rc8
Rxc8#

You explore the first path. You send two of your teammates to explore the second path. You send the remaining six teammates in groups of three to explore each of the two remaining paths. If your path leads to the exit, then everything is good. Otherwise, you ask the two groups of three whether their paths lead to the exit. If in both groups everyone answers consistently, then nobody is lying, and you will escape. If in both groups there is a person whose answer is different from the others in the group, then the majority in both groups says the truth. Once again, you will know which path leads to the exit. Finally, if in exactly one of the groups everyone answers consistently, you ask the group of two. If the team members there answer consistently with each other, then they say the truth. You will have two groups which tell the truth and will know which path leads to the exit. If the answers of the teammates in the group of two differ, then in the inconsistent group of three the majority will be saying the truth. Again, you will be able to deduce which path leads to the exit.

No, such sequence does not exist. Call a position of the stones “matching” if the stones occupy cells of the same color, and “opposite” if the stones occupy cells of opposite colors. If such sequence exists, then the number of matching and opposite positions must differ by at most 1. However, the number of matching positions is 64×31, whereas the number of opposite positions is 64×32, so this is not true.

The front wheels are not parallel to each other. The reason is that when steering, all wheels are turning along arks that have a common center. Otherwise, the car would be drifting.