Flavius Josephus’s Problem - Lycée Emmanuel d'Alzon (Nîmes)

The following game is played with $n$ players, seated in a circle. Each player is assigned a number (between $1$ and $n$).
We choose an integer $k \geq 2$. At the $m$-th step of the game, the player $mk$ is eliminated, and when we reach $n$, we return to the first remaining player.
One of the players can choose their number $n_0$. He aims at finding $n_0$ in order to be the last eliminated player.
For example, for $n = 10$ and $k = 3$, the elimination order is: $3; 6; 9; 2; 7; 1; 8; 5 ; 10; 4$.
The questions that arise:
– How to choose the number $n_0$?
– Is there a formula that allows us to directly determine the elimination order of a player i? (for example, to say directly that player $2$ will be eliminated in the fourth round)?