A072842

[Rob Pratt]

On Fri, 26 Jul 2002, Don Reble wrote:

> > %S A072842 2,8,23,52
> > %N A072842 Largest m such that we can partition the set {1,2,...,m} into
> >     n disjoint subsets with the property that we never have a+b=c for
> >     any a, b, c in any of the subsets.
> > %C A072842 The fourth number may be erroneous.
> > %e A072842 max(m(2)) = 8 because we may partition the set into
> >     {1, 3, 5, 8} and {2, 4, 6, 7} but in no other ways; attempting to
> >     add 9 to either will produce a set with the property that a+b=c for
> >     some a,b,c (1+8=9 or 2+7=9)
> > %A A072842 Tor G. J. Myklebust (pi at flyingteapot.bnr.usu.edu), Jul 24 2002
>
>     The example should be { 1 2 4 8 } { 3 5 6 7 }.
>
>     a(4) is at least 58.
> 	{ 1 2 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 }
> 	{ 3 5 6 12 20 27 41 42 56 57 }
> 	{ 8 9 11 14 15 18 21 47 48 50 51 53 54 }
> 	{ 17 23 24 26 29 30 32 33 35 36 38 39 44 45 }
>     Hmm... If it is exactly 58, my little program would need about 10^15
>     years to prove it. Don't wait up.
>
> --
> Don Reble       djr at nk.ca

a(4) is at least 66 (due to Ernst Munter):

{ 24 26 27 28 29 30 31 32 33 36 37 38 39 41 42 44 45 46 47 48 49 }
{ 9 10 12 13 14 15 17 18 20 54 55 56 57 58 59 60 61 62 }
{ 1 2 4 8 11 16 22 25 40 43 53 66 }
{ 3 5 6 7 19 21 23 34 35 50 51 52 63 64 65 }

See Dr. Dobb's Journal--Solutions to the "Monopoles" Problem
(http://www.ddj.com/documents/s=896/ddj9912m/9912m.htm).

Rob Pratt
Department of Operations Research
The University of North Carolina at Chapel Hill

rpratt at email.unc.edu

http://www.unc.edu/~rpratt/