January 2015








Solution

Everyone gave a different answer. This is important. Now, there are twelve people and the spouses do not shake hands with each other. This implies that every person can shake hands with at most ten others. Thus the maximum number of hands that can be shaken is ten and minimum number is zero.

Prakash has asked eleven people (including his wife) as to how many persons they had shaken hands with. He received eleven answers and each answer was different. This implies that the answers he had received ranged from zero to ten. Thus, there was a person who had not shaken hands with anybody, a second who had shaken hands with only one person, a third who had shaken hands with two persons, a fourth who had shaken hands with three persons, a fifth who had shaken hands with four persons, a sixth who had shaken hands with five persons, a seventh who had shaken hands with six persons, an eighth who had shaken hands with seven persons, a ninth who had shaken hands with eight persons, a tenth who had shaken hands with nine persons, and an eleventh person who had shaken hands with ten people. Thus, the eleven answers can be represented as the following set: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

For the ease of identifying the persons, let's name them by the number of persons they have shaken hands with. Thus, Zero has not shaken hands with anybody, One has shaken hands with one person, Two has shaken hands with two persons, and so on.

Let's consider the case of Ten first. Ten has shaken hands with ten people. Ten has not shaken hands with his/her spouse. This implies that the ten people who have shaken hands with Ten have shaken hands with at least one person (Ten). The only person not to have shaken hands with anybody, therefore, has to be Ten's spouse (everyone has given a different answer, so there has to be a Zero). So, we have identified one couple - Ten and Zero. Let's represent this couple as (Ten, Zero).

Let's take the case of Nine now. Nine has shaken hands with nine people. Nine has not shaken hands with his/her spouse and with Zero (Zero has not shaken hands with anybody). Thus, these nine people with whom Nine has shaken hands have shaken hands at least twice (they have already shaken hands with Ten). Who, then, is the person who has shaken hands with only one person? That person, obviously, has to be Nine's spouse. Thus, we have the second couple (Nine, One).

Let's turn our attention to Eight. Eight has shaken hands with eight people, excluding his/her spouse, Zero (Zero has not shaken hands with anybody), and One (One has shaken hands with only one person - Ten). Thus, the eight people with whom Eight has shaken hands have shaken hands at least thrice (with Ten, Nine and Eight). Thus the only person who has shaken hands with only two persons has to be Eight's spouse. Thus, we have the third couple, (Eight, Two).

Now, we come to Seven. Seven has shaken hands with seven people. Seven has not shaken hands with his/her spouse, Zero, One and Two. The seven people with whom Seven has shaken hands have, thus, shaken hands at least four times (with Ten, Nine, Eight, and Seven). Thus, the only person who could have shaken hands with only three persons is Seven's spouse. So, here we have the fourth couple (Seven, Three).

Now, we come to Six. Six has shaken hands with six people. Six could not have shaken hands with his/her spouse (spouses do not shake hands with each other), with Zero (Zero does not shake hands), with One (One has already shaken hands with Ten and is not going to shake any more hands), with Two (Two has shaken hands with Ten and Nine, and is not going to shake any more hands), and with Three (Three has completed the quota of three handshakes by shaking hands with Ten, Nine and Eight). The remaining six persons with whom Six shakes hands, thus, have shaken hands with at least five persons (with Ten, Nine, Eight, Seven and now with Six). So the only person who could have shaken hands with exactly four persons is Six's spouse. Thus, we have the fifth couple, (Six, Four).

We have, thus, determined five couples: (Ten, Zero), (Nine, One), (Eight, Two), (Seven, Three), and (Six, Four). We are left with Five and Prakash himself. Five is, therefore, Prakash's wife. So the sixth couple is (Prakash, Five). Prakash's wife has, therefore, shaken hands with five persons.

With how many persons has Prakash shaken hands with? Now, the number of handshakes by each couple sums up to ten; thus we have the couples: (Ten, Zero), (Nine, One), (Eight, Two), (Seven, Three) and (Six, Four). Thus, it would be logical to conclude that the sixth couple should be (five, Five). So, Prakash has shaken hands with five others. Let's see if this holds. Ten, Nine, Eight, Seven, and Six have definitely shaken hands with Prakash (otherwise, they could not have been able to fulfill their quotas).

Ten has shaken hands with ten people excluding his/her spouse. Thus, Ten has had to shake hands with Prakash. Similarly, Nine has shaken hands with nine people excluding his/her spouse and Zero (Zero is content with "namastes"; Zero is not shaking hands with anybody). Thus, the remaining nine people with whom Nine has shaken hands has to include Prakash. Eight could not have shaken hands with Zero (Zero is not shaking hands), One (One has already shaken hands with Ten, and is not shaking any more hands) and Two (his/her spouse); the remaining eight persons with whom Eight has shaken hands has to include Prakash. Seven could not have shaken hands with Zero, One, Two and Three (his/her spouse); the remaining seven people with whom Seven has shaken hands has to include Prakash. Six could not have shaken hands with Zero, One, Two, Three and Four (his/her spouse); the remaining six people with whom Six has shaken hands has to include Prakash.

Thus, Prakash has shaken hands with at least five persons - Ten, Nine, Eight, Seven and Six.

Prakash could not have shaken hands with Zero (Zero is content with doing a "namaste" and does not like to shake hands). He could not have shaken hands with One (One has already shaken hands with Ten and is not going to shake any more hands. He could not have shaken hands with Two (Two has shaken hands with Ten and Nine, and is not going to shake any more hands). He could not have shaken hands with Three (Three has shaken hands with Ten, Nine and Eight, and is not going to shake any more hands). He could not have shaken hands with Four (Four has shaken hands with Ten, Nine, Eight and Seven). And, of course, Prakash cannot have shaken hands with Five, his wife. So, Prakash, like a good husband, has shaken hands with five persons just as his wife.

Did I get that right? Or, was I wrong as always?



Some useful links for
your career:


  • Union Public Service Commission - www.upsc.gov.in
  • IIT-Kharagpur - www.iitkgp.ac.in
  • Indian Statistical Institute - www.isical.ac.in
  • Indian Institute of Technology Madras - www.iitm.ac.in
  • Indian Institute of Management, Ahmedabad - www.iimahd.ernet.in
  • Indian Institute of Mass Commission - www.iimc.nic.in
  • IIT Bombay - www.iitb.ac.in
  • Indian School of Mines, Dhanbad - www.ismdhanbad.ac.in
  • Birla Institute of Technology, Ranchi - www.bitmesra.ac.in
  • Central Institute of Fisheries Nautical and Engineering Training - www.cifnet.nic.in
  • Indian Institute of Information Technology, Allahabad (Deemed University) - www.iiita.ac.in
  • Central Marine Fisheries Research Institute, Kochi - www.cmfri.com
  • Tata Institute of Social Sciences, Mumbai - www.tiss.edu

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