2680 challenge

From: Risto Lankinen (rlankine_at_hotmail.com)
Date: 01/24/05 

Date: 24 Jan 2005 11:29:42 -0800

Clive Tooth wrote:
> The text file
> http://www.pisquaredoversix.force9.co.uk/2680.txt
> consists of 2680 lines. Each line contains five integers. Each of these
> integers is in the range 0 to 244. The challenge is to select 49 of the 2680
> lines in such a way that each of the numbers from 0 to 244 is present in the
> selection.
>
> I do not know if there is a solution to this challenge.

This is a well-obfuscated way of asking if 5x7x7 box can be packed with
the F-pentomino. To see what is an F-pentomino, convert any line of the
text file into base-7 and plot the (three) resultant 7:ary digits in 3D.

There are 2680 orientations of an F-pentomino within a 5x7x7 box, and a
quick (sample-based) check suggests that they are all represented in the
text file. Selecting a set of 49 non-overlapping integer lines hence is
equivalent to finding a set of 49 F-pentominoes that fill the box.

Quick scan of the net indicates that this may be an open problem in the
art of packing theory...

http://www.mathematik.uni-bielefeld.de/~sillke/PENTA/qu-prime

[See "7) F1 box 5x7x7" near the end of the article.]

 - Risto -