Put on Your Problem Solving Hat III

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dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

by dtweah » Tue May 12, 2009 6:19 am

There are 2005 people at a party. By the end of the party, in every group of four people there is
at least one person who has shaken hands with the other three members of the group. What is
the smallest possible number of people who have shaken hands with everyone at the party?

a. 1002
b. 1003
c. 2001
d. 2002
e. 2003

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Source: Beat The GMAT — Problem Solving | Tue May 12, 2009 6:19 am

avenus: Master | Next Rank: 500 Posts; Posts: 238; Joined: Tue Feb 10, 2009 8:44 am; Thanked: 9 times

by avenus » Wed May 13, 2009 4:31 pm

Suppose all but 4 people have shaken hands with everyone. If we take the remaining 4 and form a group, that group will not meet the requirement (they will have shaken hands with many out of the group but none of them will have shaken hands with the other 3 within the group). This tells us n>2001.

Now assume 1 of those socially impaired people, following the advice of his $200-an-hour shrink, decides to go wild and shakes hands with everyone at the party. We now have someone that's shaken hands with the other 3 and the group of outcasts is no longer such. n = 2002

Answer is D

Last edited by avenus on Thu May 14, 2009 7:46 am, edited 1 time in total.

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dtweah: Master | Next Rank: 500 Posts; Posts: 487; Joined: Fri Mar 27, 2009 5:49 am; Thanked: 36 times

by dtweah » Wed May 13, 2009 4:47 pm

avenus wrote:Suppose all but 4 people have shaken hands with everyone. If we take the remaining 4 and form a group, that group will not meet the requirement (they will have shaken hands with many out of that group people but none of them will have shaken hands with the other 3 within the group). This tells us n>2001.

Now assume 1 of those socially impaired people, following the advice of his $200-an-hour shrink, decides to go wild and shakes hands with everyone at the party. We now have someone that's shaken hands with the other 3 and the group of outcasts is no longer such. n = 2002

Answer is D

Brilliant. I will post the OA which to me is not too informative. I reasoned it out this way.

2005/4= 501 remainder 1. The problem says every gruop of 4 must have 1 person who has shaken hands with the rest in the group. There are 501 such groups of four. Which means any random scatterring and reassignment must meet that condition. Suppose that 2000 people are the min. Then the rest 5 people will have nobody who has shaken hands with the rest in the group . But this contradicts the conditin of the problem so it cannot be true that 2000 is the min. What about 2001? There will still be 4 people who can form a group that contradicts the hypothesis. 2002? Yes b/c the problem does not require a group of 3 only a group of 4.

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