The Flat Polycube
Touching Problem
on the integral 3Dgrid.
Copyright Karl Scherer 2005
on the integral 3Dgrid.
Copyright Karl Scherer 2005
Definitions
 A 'polycube' is a polyform based on cubes of same size, the cubes being glued together on full faces.
 A 'kcube' is a polycube made from k cubes.
 A 'flat polycube' is one which can be placed in such a way that it extends only one cube high into the third dimension. Simple examples are any 'rods' 1 x 1 x n and any 'flat boxes' 1 x n x m.
The Flat Polycube Touching Problem
a) For each n, try to find the smallest k such that a flat kcube exists with the following properties:
 if arranged appropriately in three dimensions, n congruent copies of the kcube share some surface with any other copy
 each polycube in this assembly is aligned to an integral 3dimensional grid. (If the polycubes need not be aligned to an integral grid, there is no maximum n. This is easy to see. Also, the problem becomes much less interesting.)
The author has found solutions up to n=14 (see table of results below) and is convinced that n=16 can be achieved (the tiles will be very large and complex).
b) Also, try to solve the same problems for two layers (the third dimension has thickness 2).
Here the maximum n is 8. The author has found solutions for all n = 1,...,8, see second table below.
Table of results for problem part a)
The solutions presented here might not be the best possible. Can you improve on them?
Table of results for problem part b)
The solutions presented here might not be the best possible. Can you improve on them?
n 
k 
Images of the polycube and the assembled set.  Author 
Comment 

12 
1 
1x1x1 cube 

34 
2 

1x1x2 rod 

5 
4 
KS 
five Ltetracubes 

6 
5 
The two layers assembled separately Final assembly 
Erich Friedman 2005 
six pentacubes 
7 
12 
The two layers assembled separately. All seven pieces assembled Perspective view 
KS 2005 
seven 12cubes 
8 
39 
The 39cube which solves the problem The two layers assembled separately. All eight pieces assembled Perspective view  Here are two more, similar solutions: 
KS 2005 
eight 39cubes 
All images drawn by the author with the marvellous 3D drawing program Rhinoceros 1.1.