Suppose that in a sports tournament featuring $n$ players, each pair plays one game and there is always a winner and a loser (no draws). Show that the players can be arranged in an order $P_1, P_2......P_n$ such that player $P_i$ has beaten $P_{i+1}$ for all $i = 1,2,....,n-1$
PS:-I tried but I cannot understand how to do the problem in a mathematical way.
PPS:- This is the $Q.1$ of $ISI$ entrance exam $2016$.
Show that $n$ players can be arranged in an order $P_1, P_2......P_n$ such that
0
$\begingroup$
sequences-and-series
-
0You are looking for a [topological sort](https://en.wikipedia.org/wiki/Topological_sorting) – 2017-02-12
-
0@rogerl meaning? – 2017-02-12
-
0@rogerl topological sorting is beyond the course of ISI – 2017-02-12
1 Answers
2
Induction works. For base case, it's clearly true for one or two players.
Say it's possible for $k$ players no matter the result. Take a $k+1$ player tournament, let the players play, choose $k$ of them and order them as prescribed. Now, either the unplaced player won or lost against $P_1$. If he won, then make him $P_0$, and you're done. If he lost, look at $P_2$. If he won against $P_2$, insert him between $P_1$ and $P_2$. If not, continue to $P_3$, and so on. Lastly, if he lost all his games, put him behind $P_k$.