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 The odds stay unchanged (Posted on 2004-06-17)
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 ½a(a-1) for one club and ½b(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).

 Subject Author Date Puzzle Solution K Sengupta 2008-05-27 15:50:58 Answer K Sengupta 2008-05-23 06:14:12 full solution Bon 2004-07-12 17:24:56 che?? vije 2004-07-07 13:22:37 re: generalisation ==> author's remarks Ady TZIDON 2004-06-20 11:00:07 More about generalization: Gamer 2004-06-19 18:27:45 Another Approach Gamer 2004-06-18 23:22:00 re(2): ution - NON trivial solution Ady TZIDON 2004-06-17 18:23:25 re: Non-trivial answer by brute force Dan Blume 2004-06-17 16:28:30 re: Non-trivial answer by brute force - sorry, no Thalamus 2004-06-17 16:25:12 Non-trivial answer by brute force Dan Blume 2004-06-17 16:18:08 re(2): A trivial solution - NON trivial solution SilverKnight 2004-06-17 15:51:30 re: A trivial solution - NON trivial solution Erik O. 2004-06-17 15:41:35 A trivial solution Erik O. 2004-06-17 15:13:39 hints - partial solution Erik O. 2004-06-17 14:41:36 re(2): First meeting (hint) Dan Blume 2004-06-17 14:00:50 re: Possibility? (spoiler?) Charlie 2004-06-17 13:57:38 re: Possibility? (spoiler?) SilverKnight 2004-06-17 13:57:26 Possibility? (spoiler?) Richard 2004-06-17 13:54:38 re: First meeting (hint) Thalamus 2004-06-17 13:54:02 First meeting Dan Blume 2004-06-17 13:45:24

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