Rehm's 3-Cubes (1980): Torsten Sillke, 2000-12-06 ---------------------- Building a 3-cubes with different tetracubes and tricubes. There are 2 differnt tricubes. Both fit into the 3-cube. There are 7 differnt tetracubes which fits into a 3-cube (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 3-cube. 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, p189-190, 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 3-cube 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 3-cube. This shows that the 3-cube 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 2000-12-06. 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) Urania-Verlag Leipzig, Jena, Berlin, 2. Aufl. 1989 (1. Aufl. 1988). ISBN 3-332-00116-7. - Somasolos No. 3: Rehm's cubes p176, p189-190 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 3-332-00480-8. - 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 (1919-1995) a teacher of mathematics on Realschule Herzberg http://home.t-online.de/home/Realschule.Herzberg/ -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/