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.

Topic
Combinatorics

Difficulty Level

7 out of 10

Suggested Book

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 

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Similar Problem

Consistency and MVUE |ISI MStat PSB 2006 Problem 9

This is a very simple sample problem from ISI MStat PSB 2006 Problem 9. It’s based on point estimation and finding consistent estimator and a minimum variance unbiased estimator and recognizing the subtle relation between the two types. Go for it!

Balls-go-round |ISI MStat PSB 2013 Problem 10

This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 10. It’s based mainly on counting and following the norms stated in the problem itself. Be careful while thinking !

ISI MStat PSB 2005 Problem 5 | Uniformity of Uniform

This is a simple and elegant sample problem from ISI MStat PSB 2005 Problem 5. It’s based the mixture of Discrete and Continuous Uniform Distribution, the simplicity in the problem actually fools us, and we miss subtle happenings. Be careful while thinking !

ISI MStat PSB 2012 Problem 2 | Dealing with Polynomials using Calculus

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 2 based on calculus . Let’s give it a try !!

ISI MSTAT PSB 2011 Problem 4 | Digging deep into Multivariate Normal

This is an interesting problem which tests the student’s knowledge on how he visualizes the normal distribution in higher dimensions.

ISI MStat PSB 2012 Problem 5 | Application of Central Limit Theorem

This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 5 based on the Application of Central Limit Theorem.

ISI MStat PSB 2007 Problem 7 | Conditional Expectation

This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 7. It’s a very simple problem, which very much rely on conditioning and if you don’t take it seriously, you will make thing complicated. Fun to think, go for it !!

ISI MStat Entrance Exam books based on Syllabus

Are you preparing for ISI MStat Entrance Exams? Here is the list of useful books for ISI MStat Entrance Exam based on the syllabus.

ISI MStat PSB 2008 Problem 8 | Bivariate Normal Distribution

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 8. It’s a very simple problem, based on bivariate normal distribution, which again teaches us that observing the right thing makes a seemingly laborious problem beautiful . Fun to think, go for it !!

ISI MStat PSB 2004 Problem 6 | Minimum Variance Unbiased Estimators

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 6. It’s a very simple problem, and its simplicity is its beauty . Fun to think, go for it !!