in reply to Lunch Bunch arrangement problem

Did you run across a proof that there are no lunch bunch arrangements for orders that arent powers of one prime? Basically this would be a generalization of the 36 officer problem. It seems pretty likely that your statement is correct

I think it's impossible for composite orders like 6 or 10, but all orders that are powers of primes should work.

Seeing as those are the only orders of Galois fields.
I'm asking because if you're not aware of a generalized proof, I'd probably be willing to take a shot at it. I havent done much algebra in a number of years, but it would be a good refresher. :)

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Re: Re: Lunch Bunch arrangement problem
by tall_man (Parson) on May 21, 2003 at 19:09 UTC
    I don't know of a general proof. I know Euler had a conjecture that two orthogonal Euler squares of order 10 could not be found. That conjecture was disproved when a 10x10 pair was found (but a pair is not enough to create lunch bunches). Here is the reference.

    Update: A lunch bunch arrangement is a resolvable block design of the form (n^2, n, 1), which is also known as an affine plane. An affine plane of order n exists if and only if a projective plane of order n exists. In the projective plane article it states:

    A finite projective plane exists when the order n is a power of a prime...It is conjectured that these are the only possible projective planes, but proving this remains one of the most important unsolved problems in combinatorics.

    So if one could prove the above conjecture, it would prove that the only lunch bunches are of the order of powers of a prime.

[reply]
      Well, with that knowledge, this is probably beyond the scope of my time and / or ability. But we'll see...
[reply]

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