# Understand the problem

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 P1, P2, . . . , Pn such
that player Pi has beaten Pi+1 for all i = 1, 2, . . . , n.

##### Source of the problem

I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016. Subjective Problem no.1.

Combinatorics

7 out of 10

# Start with hints

Do you really need a hint? Try it first!

we define a sequence $$p_1$$>$$p_2$$>$$p_3$$………..>$$p_n$$ . such that $$p_i>p_{i+1}$$ implies $$p_i$$ has beaten $$p_{i+1}$$

let assume that the statement is true for n=k i.e $$p_1> p_2>p_3………p_k$$ now look into the case of $$p_{k+1}$$

if $$p_{k+1}$$ > $$p_1$$ or $$p_k$$>$$p_{k+1}$$ we are done (why?)

if $$p_k$$<$$p_{k+1}$$ or $$p_{k+1}$$<$$p_1$$ then we can find out at least one pair $$p_i$$ >$$p_{i+1}$$ such that $$p_i$$> $$p_{k+1}$$ >$$p_{i+1}$$ , we can just find it by shifting $$p_{k+1}$$ term by term [ you can make some small experiment ]. other wise it contradict our initial assumption that $$p_k$$<$$p_{k+1}$$ or $$p_{k+1}$$<$$p_1$$ [ why?] hence we are done

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