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        riddles >> easy >> sum of N real numbers
        
(Message started by: BNC on May 11^th, 2003, 12:36am)

Title: sum of N real numbers
Post by BNC on May 11^th, 2003, 12:36am

The sum of N real numbers (not necessarily unique) is 20.
The sum of the 3 smallest of these numbers is 5. The sum of the 3 largest is 7.

What is N?

Title: Re: sum of N real numbers
Post by LZJ on May 11^th, 2003, 12:46am

N = 10 ? Wild guess :-*

Title: Re: sum of N real numbers
Post by harpanet on May 11^th, 2003, 8:19am

I like this one. You start off thinking (well I did, anyway :D) that there could be any number of possible answers, after all we're dealing with Real numbers. But when you start to analyse it you see how restrictive the conditions are.

Title: Re: sum of N real numbers
Post by towr on May 11^th, 2003, 8:25am

9 or 10

Title: Re: sum of N real numbers
Post by Icarus on May 11^th, 2003, 11:36am

How can you do it with N=9, towr?

Here is my analysis: [hide]The remaining numbers add up to 8. The bottom three numbers must have a least one entry >= their average of 5/3. The top three numbers must have one entry less than their average of 7/3. Thus all the remaining numbers x must satisfy 5/3 <= x <= 7/3. Let K = N-6 be the count of remaining numbers. Then K(5/3) <= 8 and K(7/3) >= 8, so 24/7 <= K <= 24/5. Since K is an integer, K = 4 and N = 10.[/hide]

Title: Re: sum of N real numbers
Post by towr on May 11^th, 2003, 3:14pm

on 05/11/03 at 11:36:34, Icarus wrote:

How can you do it with N=9, towr?

I don't think I can, come to think of it..
I made some bad errors..

Title: Re: sum of N real numbers
Post by Bozer on Dec 16^th, 2003, 6:46am

N = 10
The whole set:
{1, 2, 2, 2, 2, 2, 2, 2, 2, 3} has 10 members

subset of the three smallest:
{1 + 2 + 2} sum to 5

subset of the three largest:
{2 + 2 + 3} sum to 7

remaining memebers of the set:
{2 + 2 + 2 + 2} sum to 8

Title: Re: sum of N real numbers
Post by rmsgrey on Dec 16^th, 2003, 8:25am

Well, that's the unique integer solution...

The problem specified real numbers, so there are infinitely many possible solutions, all of which must have ten numbers (as shown by Icarus)

Title: Re: sum of N real numbers
Post by koolking on Jun 8^th, 2012, 9:05pm

I want to check the answer if its 10
Let me give my thinking behind it
If we assume the nos are integers then the answer is given in earlier posts
Let us take the other case
sum of largest 3 nos are 7 so the no smallest f them is less or equal to than 7/3. similarly the largest of the smallest 3 nos is >=5/3
now the sum of rest of the nos is 8
so take the 4th smallest no which is >= 5/3 so atmax we need 4.8 times of this no to make 8
also take 4th largest no min of 3.5 nos is needed of this to make 8
all the nos are between 5/3 and 7/3 so answer is we need 4 nos to make 8 and hence making of total 10 nos

Title: Re: sum of N real numbers
Post by atyq on Jul 27^th, 2012, 5:20pm

{1, 2, 2, 2, 2, 2, 2, 2, 2, 3} has 10 members.
You can verify by yourself: lit works.

Title: Re: sum of N real numbers
Post by littlemisschic on Jul 29^th, 2012, 5:35pm

N=2.5?

another wild guess but it seems to fit if my calcs are correct!

Title: Re: sum of N real numbers
Post by SMQ on Jul 30^th, 2012, 5:55am

on 07/29/12 at 17:35:13, littlemisschic wrote:

N=2.5?

another wild guess but it seems to fit if my calcs are correct!

Can you provide an example of a list with two-and-a-half numbers in it? ;)

--SMQ

Title: Re: sum of N real numbers
Post by peoplepower on Jul 30^th, 2012, 6:53pm

on 07/30/12 at 05:55:50, SMQ wrote:

Can you provide an example of a list with two-and-a-half numbers in it? ;)

--SMQ

Easy, {11,11,1}

Title: Re: sum of N real numbers
Post by manchester121 on Nov 4^th, 2014, 12:00am

sum=(n(n+1))/2
put n valve and find sum of real no...easily...

Title: Re: sum of N real numbers
Post by csendra on Nov 5^th, 2014, 11:32pm

on 12/16/03 at 06:46:53, Bozer wrote:

nice work and the yes right is the 10 i,m agree with it