Math Is Fun Forum

samuel.bradley.99

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Registered: 2016-03-16

Posts: 51

Handshakes

50 people are invited in a party. Each of them shook hands with at least one other guest. Assuming there was a total of 500 handshakes exchanged and that each pair of guests shook hands only once, what is the maximum number of people who shook hands with all other guests?

thickhead

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Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

samuel.bradley.99

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Registered: 2016-03-16

Posts: 51

Re: Handshakes

Solution please??

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Hi samuel.bradley.99;

A bit wordy...someone else may be able to put it more concisely.

Last edited by phrontister (2016-04-25 14:45:18)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Oh...I overlooked that.

I think your answer is right, but I haven't checked your method yet. Gotta go out now for a couple of hours at least, and I'll look at it again later.

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

anna_gg

Member

Registered: 2012-01-10

Posts: 232

Re: Handshakes

Yes but you must add also the 50 initial handshakes, since we know that each invitee shook hands with at least one other. So we start with 50 and then the first person that shook hands with EVERY other, had 48 more (since we have already counted the first one), the second 47 etc.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

Last edited by thickhead (2016-04-25 18:16:49)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

anna_gg

Member

Registered: 2012-01-10

Posts: 232

Re: Handshakes

Ohh yes, you are right! I am stupid

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Hi thickhead;

Yes, your method works fine!

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

First 11 people who handshook with all others exhausted 484 handshakes. 12th man at best can shake hands with 16 more along with 11 he has already completed bringing his total to 27.What about the remaining? 16 people would have 12 handshakes and 22 would have 11 handshakes each. I think this competes the solution.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Hi;

Here's a graphic solution:

Last edited by phrontister (2017-02-23 11:42:36)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

Now the original problem being over, what is the maximum number of people who can have single handshake, the total number of handshakes remaining 50?

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Hi thickhead;

That is a trickily worded question!

Still wrestling with the wording, though...

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

What I mean to say is that some people after handshaking once will retire and do not participate further in handshaking. If too many people do so the total number of handshakes will fall short of 500.e.g. If 20 people take single handshake say for simplicity between them only they have exhausted 10 handshakes and the remaining 490 handshakes will have to be completed by remaining 30 people. But 30 people can have a maximum of only

handshakes. So the number of retirees has to be less than 20. An equation or rather

type of equation has to solved.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

I'm not sure if I've got your meaning right. Is it this?...

   From a group of 50 people, what is the maximum number who only have a single handshake, so that the total number of handshakes by the group exceeds 500 by no more than 50?

If so, then my answer is this:

Last edited by phrontister (2016-04-27 11:46:24)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

I am sorry for the mistake.I meant the total number of handshakes remaining 500 as before. even when you had mentioned I had not looked back. Now only i realized it.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Ok, thanks. I think I have the picture now.

Last edited by phrontister (2016-04-27 13:11:39)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,877

Re: Handshakes

Those equations work, but they also work for smaller values of r.

Is there a notation that could be incorporated in the equation to indicate that maximum r is required, or should that be worded somehow with the answer, or don't we worry about putting it in the answer because it's in the question?

Also, is there some way of arriving at a solution without testing several values of r to find out which is the maximum? I suppose you could solve for r where the rhs = 500, and round the answer down to the nearest even integer. Btw, my little HP 32SII calculator's Solve function gives the answer r = 18.1617121098.

In my previous post I had this equation:

That came from this:

I should have included the inequality:

Last edited by phrontister (2016-04-28 12:33:21)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Handshakes

We can think of maximum no. of people with zero handshake but it becomes a routine and not worth going for.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}