On 10/3/2012 9:52 PM, Fred lunnon wrote:
Nice problem.
Can't answer 2) at this stage, though strongly suspect yes!
I'd have to agree.
My best so far non-P set of 8 is
{86,85,84,82,79,73,62,42}
Can anyone improve on 86?
This 1988 paper by one W. F. Lunnon shows that 84 is optimal: http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917837-5/ (The definition there omits "non-empty" and "disjoint", but that doesn't actually change the property: if two distinct overlapping subsets have the same sum, so do the disjoint subsets produced by removing their intersection.)
On 10/3/12, Victor Miller <victorsmiller@gmail.com> wrote:
Say that a finite set of integers, A, has *property P* is there are two non-empty disjoint subsets S and T of A, such that sum(S) = sum(T) (where sum(X) is the sum of the elements of X).
1) Show that every 10 element subset of {1,...,100} has property P (this is fairly easy). 2) Is it true that every 9 element subset of {1,...,100} has property P?
-- Fred W. Helenius fredh@ix.netcom.com