Question 1207618
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I run a book club with n people, not including myself. Every day, for 400 days, 
I invite 2 members in the club to review a book. What is the smallest positive integer n 
so that I can avoid ever having the exact same group of 2 members over all 400 days?
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                It looks to be complicated.

                But in reality,  it is as simple as a cucumber,  and,  in addition,  it is charming.

                When you learn it out,  you will gasp . . . 



<pre>
Let the number of the members be n.

Then the number of pairs is  {{{(n(n-1))/2}}}.


They want you find the minimum possible n such that  

    {{{(n*(n-1))/2}}} >= 400.    (1)


So, all you need is to solve this inequality.


Multiply both sides by 2

    n*(n-1) >= 2*400 = 800.    (2)


Square root of 800 is  28.2842...


So, your number n is the first integer positive number 
which satisfies (2), and it is somewhere close to 28.


Check n= 28: 28*(28-1) = 28*27 = 756  <<<---=== not enough

Check n= 29: 29*(29-1) = 29*28 = 812  <<<---=== just enough


<U>ANSWER</U>.  You should have at least 29 members in the club to make it possible.
</pre>

Solved.



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Re-phrasing O'Henry, the famous American writer,


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The blind begin to walk and the dumb begin to see 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;when they receive such beautiful solutions to their problems.