The Flat Polycube Touching Problem
on the integral 3D-grid.
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 'k-cube' 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 k-cube exists with the following properties:
  •  if arranged appropriately in three dimensions, n congruent copies of the k-cube share some surface with any other copy
  • each polycube in this assembly is aligned to an integral 3-dimensional 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.)
For which n is there a solutions at all?
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?

n
k
Images of the polycube and the assembled set. Author
Comment
1-2
1
 1x1x1 cube


3-4
2




1x1x2
rod
 5
4
KS
L-tetracube
 6
5



KS
V-pentacube
 7
6

4 pieces assembled assembled

KS
seven hexacubes
 8
6

 

 

6 pieces assembled


  
all 8 pieces assembled
Erich Friedman 2005
Eight hexacubes.
This 6-cube is also  solves the case n=7.
 9
11




KS 2005
nine
11-cubes
10
?



11
?



12
16

 





Layers:
(A=light blue, B=green, C= purple, D=yellow, E=blue, F=orange, G=red, H=pink, I=gold, J=violet, K=white, L=grey)
----------  ----------  ----------  ----I-----  ----IL----
----------  ----------  ----------  ----I-----  ---CCC----
----------  ----------  ----------  ----IK----  ---CJK----
----------  -----K----  -----K----  ----IK----  --CCCCCCCC
----GGG---  ----IKG---  ---EEEG---  -EEEIKG---  DECE--GDGC
---FFF----  ---FIK----  ---FHHH---  ---FIKHHH-  DFCF--HDHC
----------  ----I-----  ----I-----  ----IK----  DDDDDDDD--
----------  ----------  ----------  ----IK----  ----ILD---
----------  ----------  ----------  -----K----  ----DDD---
----------  ----------  ----------  -----K----  ----JK----


----IL----  -----L----  ----------  ----------  ----------
----AAA---  -----L----  ----------  ----------  ----------
----JKA---  ----JL----  ----------  ----------  ----------
AAAAAAAA--  ----JL----  ----J-----  ----J-----  ----------
AEBE--GAGB  ---EJLGGG-  ---EGGG---  ---EJL----  ---EEE----
AFBF--HAHB  -FFFJLH---  ---FFFH---  ----JLH---  ----HHH---
--BBBBBBBB  ----JL----  -----L----  -----L----  ----------
---BIL----  ----JL----  ----------  ----------  ----------
---BBB----  ----J-----  ----------  ----------  ----------
----JK----  ----J-----  ----------  ----------  ----------






KS 2005
twelve 16-cubes

Note that the center of the assembly contains an empty 2x2x2 cube-shaped space.
14
119

...that was the rough idea...


that's the monster tile I found which solves the problem...
(size reduced to 119 on July 1, 2005)

..3 pieces assembled...


...6 pieces assembled (detail)...


10 pieces assembled


14 pieces assembled (detail)

14 pieces assembled, final view.

KS 2005
fourteen
191-cubes

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
1-2
1
 1x1x1 cube


3-4
2




1x1x2
rod
 5
4

 

The two layers, assembled separately


Final assembly

KS
five
L-tetracubes
 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
12-cubes
 8
39

 
The 39-cube which solves the problem



T
he two layers assembled separately.
 
All eight pieces assembled


Perspective view
---------------------------------------------------------------------------
Here are two more, similar solutions:



KS 2005
eight
39-cubes


All images drawn by the author with the marvellous 3D drawing program Rhinoceros 1.1.