The First Algorithm is the solution to the Josephus Problem.
The problem is based on a tale during the Roman Empire, 41 jews were hiding in a cave and were about to be captured by the Romans but the jews preferred suicide over surrender and hence devised a strategy. They all would stand in a circle and every third person would be killed. Josephus did not want to commit suicide and as he was a smart man he calculated the exact position where he should stand so he is the last man standing.
The Modern problem is based on a similar lines but instead of 41, n people are given and instead of every third person to be killed every k person is killed. You have to find the safe position
Say n = 5 and k = 2, then the order of killing is 2,4,1,5
Here we observe that after the first kill i.e the killing of second person we have n-1 people and so the problem can be considered as finding the safe position with n-1 people when person on k position is executed. So, the problem has a nice recursive structure but the only problem is after the first kill the k+1 position person assumes the first position which can be adjusted as \[k \% n + 1\]
1 2 3 4 5
For a nice visualisation of the Josephus Problem visit the following link: Josephus