A group of k villagers must cross, the Sarawak river, which is a wide and deep river. There is no bridge in sight. They notice two 10-year-old boys playing in a rowboat at the shore. The boat is so tiny and it can hold only two boys or one villager. How can the villagers can get across the river and leave the boys in joint possession of the boat? How many times the boat pass from shore to shore? [Hint: Solve the problem by starting with k = 1]

Assume the boat and the boys are on the same side of the river at the start and finish.

To get one villager across, the two boys must row over, leave a boy on the other side, row back alone, let a villager row over, then then boy on the other side will bring the boat back, completing one cycle. The two boys are on the initial side of the river.

So it takes two return trips for one villager to get over.

But the question above state that a group of k villagers. can we assume that only 1 villager row over?

To solve this problem, let's start by considering the case when there is only one villager (k = 1). Since the boat can only hold two boys or one villager, the villager can simply use the boat to cross the river alone.

Now, let's move on to the case with two villagers (k = 2). One villager can take the boat to the other side, leaving one villager behind. The villager who crossed then takes the boat back to pick up the other villager. Both villagers are now on the other side, and the boat is back where it started. We can see that the boat passes from shore to shore twice in this case.

Now, let's consider the case with three villagers (k = 3). The first villager takes the boat to the other side and leaves the boat behind. The second and third villagers then cross the river together using the boat. The second villager drops off the boat on the starting side and returns alone to pick up the first villager. The first villager and the second villager both cross the river using the boat, leaving the first villager on the other side. Finally, the second villager takes the boat back to the starting side, completing the process. We can see that the boat passes from shore to shore five times in this case.

By analyzing further, we can see that for any number of villagers (k), the boat will pass from shore to shore (2k - 1) times. This means that the boat will pass the river 2k - 1 times to ensure that all the villagers cross and the boys are left in joint possession of the boat.