野菜花' problem posted at WXC

fzy · 发表于 2005-3-30 18:21:37

8位歌手参加艺术节，为他们安排 M 场演出。每场由其中 4 位登台表演，要求 8 位歌手中任意两位同时演出的场次一样多，设计一种方案，使演出的场数 M 最小。

The problem was posted by, well, by 野菜花. The poster said she is not the real 野菜花, it turns out that the account was closed and the screen name is available. So I dont know if you (the real 野菜花) want to do something about it.

fzy · 发表于 2005-3-31 22:45:08

I was not trying to persuade you to go back to WXC. Although I would love to see it, I accepted you explanation. But you could go there from your home computer to claim the screen name and put a password so that other people would not be able to do it.

Now the problem. Anybody care to solve it? Or this generalization: Suppose there are n performers performing in groups of k (n > k >= 3). What is the minimum performances so that the number of joint performances by any two performers is a constant?

Sorry I generalized your problem again. But this time I do not know the answer. I have a conjecture but cannot prove it.

fzy · 发表于 2005-4-1 18:40:54

and it is obviously a lower bound. But I cannot prove it is always achievable. (8, 4) is easy. Then I tried (7, 4), which is less easy:

ABCD, ABEF, ACEG, ADFG, BCFG, BDEG, CDEF

At that time I realized that for (15, 3) this is the Kirkman Schoolgirl problem. So I gave up.

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野菜花' problem posted at WXC

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That is my conjecture