Rehm's 3Cubes (1980): Torsten Sillke, 20001206

Building a 3cubes with different tetracubes and tricubes.
There are 2 differnt tricubes. Both fit into the 3cube.
There are 7 differnt tetracubes which fits into a 3cube (the I4 is too long).
As one tricube and six tetracubes have a total of 27 cubes
we have 2 times 7 possible sets of pieces to build a 3cube.
Frank Rehm [+] found in 1980 solutions for all combinations
with one exception and comunicated his findings R. Thiele.
He proved that the last case (removing L3 and T4) [*] is impossible.
The following table shows the number of possibilities to build
the cube if you remove the piece shown in the row and shown in
the column. Computations by T. Sillke (1992).
The Soma subset is in first entry in the table.
1 1 1 1 1 1 2 2 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1
++++++++

1 1 1  240 39 47 221 337 337 261
+
1 1 
1  138 27 no 99 245 245 31
+
If the set of pieces is invariant under reflection (each piece has its mirror
piece) the group of reflections is used to count the number of different
solutions. Otherwise the group of rotations is used.
Thiele's Impossebility proof of [*]:
after [Thiele, Haase, 1988, p189190, Somasolos No. 3].
The proof have two parts the cornerness and the parity.
Parity: checker the cube in black and white. Let the
corner cubes be black. Then the number of black cubes
of the 3cube is even. There are 5 even tetracubes
having an even number of black cubes and one odd
tetracube having an odd number of black cubes.
Therefore the number of black cubes of the tetracubes
is odd. We conclude that the number of black cubes of
the tetracube, the I3, must be odd.
Cornerness: The L4 can match at most two corner cubes.
The other tetracubes at most one. The tricube must have
an odd number of black cubes and matches no corner cube.
Therefore we can match at most seven of the eight corner
of the 3cube.
This shows that the 3cube is impossible.
[+] Frank Rehm from Germany lived in 1980 in Schoenebeck
a village 10km south of Magdeburg on the Elbe river.
This wrote me R\"udiger Thiele on 20001206.
Reference:
 Richard Mischak;
W\"urfelspielereien,
http://www.zahlenjagd.at/wuerfel.html
(collection of cube puzzles including Rehm's cubes.)
 R\"diger Thiele, Konrad Haase;
Teufelsspiele,
(engl: plays of the devil)
UraniaVerlag Leipzig, Jena, Berlin, 2. Aufl. 1989 (1. Aufl. 1988).
ISBN 3332001167.
 Somasolos No. 3: Rehm's cubes p176, p189190
Shows a solution in all possible cases and gives an impossibility
proof for the reaming one.
 R\"udiger Thiele, Konrad Haase;
Der verzauberte Raum, Spiele in drei Dimensionen,
1991, Urania Verlag
ISBN 3332004808.
 Rehm's cubes p69
 Alpha (Mathematische Sch\"ulerzeitung),
Sonderheft: Spiele und Mathematik
[english: special edition games and mathematics]
Alpha, Vol. 31 (August 1997)
Verlag Reinhardt Becker in Velten (since 1994)
 Polyominoes, Soma Cube, Herzberger Quader
The "Herzberger Quader" (2x4x5 box with the dicube, tricubes, tetracubes)
was invented by Gerhard Schulze (19191995) a teacher of mathematics
on Realschule Herzberg http://home.tonline.de/home/Realschule.Herzberg/

mailto:Torsten.Sillke@unibielefeld.de
http://www.mathematik.unibielefeld.de/~sillke/