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Please guide me on what conclusions can I draw from the following information

Nine people are standing in a $3\times3$ matrix arrangement. Rajat is the tallest among the shortest persons in each row. Sanket is the shortest among the tallest persons in each column

A. Rajat is the 3rd shortest person

B. Sanket is shorter than Rajat

C. Sanket is the 3rd tallest person.

D. Sanket is taller than Rajat

The only thing I was able to think of was writing

Rajat${}= \max( \min(a_{11},a_{12},a_{13}) , \min(a_{21},a_{22},a_{23}) , \min(a_{31},a_{32},a_{33}) )$

Sanket${} = \min( \max(a_{11},a_{21},a_{31}) , \max(a_{12},a_{22},a_{32}) , \max(a_{13},a_{23},a_{33}) )$

solution

user3653788
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1 Answers1

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Let's assume all $9$ people are of different lengths, and let's represent their lengths by $1$ through $9$ ($9$ being the tallest)

Now consider the following arrangement:

\begin{array}{ccc} 7 &1&4\\ 5&8&2\\ 3&6&9\\ \end{array}

With this arrangement we find that Rajat is of length $max(min(7,1,4),min(5,8,2),min(3,6,9))=max(1,2,3)=3$ and Sanket is of length $min(max(7,5,3),max(1,8,6),max(4,2,9))=min(7,8,9)=7$, and so claims A,C, and D are true and B is false.

Now consider the following arrangement:

\begin{array}{ccc} 1&7&5\\ 2&8&3\\ 6&4&9\\ \end{array}

For this arrangement we find that Rajat is of length $max(min(1,7,5),min(2,8,3),min(6,4,9))=max(1,2,4)=4$, and Sanket of length $min(max(1,2,6),max(7,8,4),max(5,3,9))=min(6,8,9)=6$. So, this time, claims A and C are false, but B is still false, and D is still true.

The logic of the solution is correct: D is always true, and therefore B is always false. But as the two scenarios above show, A and C can be either true or false.

Bram28
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  • Thanks for answering. Please tell me which assumption is wrong in the solution. Please check, I have edited the question. – user3653788 Dec 09 '17 at 22:36
  • @user3653788 My apologies! For my second scenario I said that Sanket was of length $5$, but I had that wrong! And in fact, the logic in the solution is correct: it has to be the case that Sanket is taller than Rajat. I'll update my post. – Bram28 Dec 10 '17 at 00:14