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At a social bridge party every couple plays every other couple exactly once. Assume there are no ties.

  1. Ifcouples participate, prove that there is a “best couple” in the following sense: A couple uis “best” provided that for every couple v, ubeats vor ubeats a couple that beats v.
  2. Show by an example that there may be more than one best couple.

Short Answer

Expert verified

(a) We proved that if n couples participates, a couple u is best provided that for every couple v,u beats v or u beats a couple that beats v.

(b) We proved there may be more than one best couple.

Step by step solution

01

(a) Step 1: To prove u beats v or u beats a couple that beats 

Given that there are n couples participants.

Consider that u beats v.

Then if ubeats mdifferent v then we need to prove that u beats m + 1.

If u does not beat m + 1 , then one of the m v ' swill beat .

Suppose that it does not hold.

Then m + 1 vwill beat m v ' s.

Therefore, m + 1 will beat u.

Hence, if n couples participates, a couple uis best provided that for every couple v, u, beats v or u beats a couple that beats v.

02

(b) Step 2: To prove that there may be more than one best couple

Consider that u beats v , it means base condition holds.

For odd number n , all the couple participants can be the best couples, since first couple beats the second couple or the couple that beats second couple.

Also, for even number n, all the couple participants can be the best couples since first couple beats the second couple or the couple that beats second couple.

Hence, this may be possible that there may be more than one best couple.

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