More on the 2680 challenge

Sunday I blogged about the pathalogical 2680 challenge. Risto Lankinen identified the true nature of the problem as:

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...

In case (like me) you don't know what an F-pentomino is, the Wikipedia entry on Pentaminoes has explanations and images.

The original poster, "The Last Danish Pastry," confessed that Risto had spotted what he was really asking, and that he'd hoped that posing the question in the way he did might lead to the discovery of new approaches to this open problem in packing theory.

—Michael A. Cleverly
Tuesday, January 25, 2005 at 18:59