All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
The odds stay unchanged (Posted on 2004-06-17) Difficulty: 4 of 5
On a certain island each of the inhabitants is a member of one of the two existing clubs.

The membership distribution is such that when two random people meet, the probability of those two belonging to the same club is equal to the probability of them belonging to distinct clubs.

When 100 newcomers arrive on the island and each enrolls in one of the two clubs, the distribution still retains this feature. How many people belong to either club?

  Submitted by Ady TZIDON    
Rating: 4.2727 (11 votes)
Solution: (Hide)
Let us denote by a the number of members in one club, by b in the other. The total number of all possible matches consisting of the members of different clubs is thus ab, while the total number of all possible couples matching members of the same club is a(a-1) for one club and b(b-1)for the other.

Hence 2ab= a*(a-1)+b(b-1)
a²+b²-2*a*b=a+b
a+b=(a-b)²

i.e the number representing the population of the island is a square of an integer, say n². So is this number increased by 100, say m².

All we have to do is to find integer solutions for m²-n²=100, or (m+n)(m-n)=100.

The factorization of 100 should be made into two even factors, to assure integer m and n. Two possible solutions are 10*10 or 50*2. One set of equations gives n=0, m=100 (a called trivial solution ), and the n=576, m=676.

Now one can go back and find the appropriate a and b. For the trivial solution 100 equals a+b and 10=a-b (45 and 55 members), and for the non-trivial solution 576 equals a+b and 24=a-b (276 and 300 members), 676 equals a+b and 26=a-b (325 and 351 members).

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionPuzzle SolutionK Sengupta2008-05-27 15:50:58
AnswerK Sengupta2008-05-23 06:14:12
Solutionfull solutionBon2004-07-12 17:24:56
Some Thoughtsche??vije2004-07-07 13:22:37
Hints/Tipsre: generalisation ==> author's remarksAdy TZIDON2004-06-20 11:00:07
More about generalization:Gamer2004-06-19 18:27:45
Another ApproachGamer2004-06-18 23:22:00
re(2): ution - NON trivial solutionAdy TZIDON2004-06-17 18:23:25
re: Non-trivial answer by brute forceDan Blume2004-06-17 16:28:30
re: Non-trivial answer by brute force - sorry, noThalamus2004-06-17 16:25:12
Non-trivial answer by brute forceDan Blume2004-06-17 16:18:08
re(2): A trivial solution - NON trivial solutionSilverKnight2004-06-17 15:51:30
Solutionre: A trivial solution - NON trivial solutionErik O.2004-06-17 15:41:35
SolutionA trivial solutionErik O.2004-06-17 15:13:39
hints - partial solutionErik O.2004-06-17 14:41:36
re(2): First meeting (hint)Dan Blume2004-06-17 14:00:50
re: Possibility? (spoiler?)Charlie2004-06-17 13:57:38
re: Possibility? (spoiler?)SilverKnight2004-06-17 13:57:26
Hints/TipsPossibility? (spoiler?)Richard2004-06-17 13:54:38
Hints/Tipsre: First meeting (hint)Thalamus2004-06-17 13:54:02
Some ThoughtsFirst meetingDan Blume2004-06-17 13:45:24
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information