Torsten Sillke, FRA, 1997-05-14 High symmetric 5*5 figures with the 'Baumeisterspiel' set: ---------------------------------------------------------- This puzzle was invented in the 1960 by H. C. Lange (Elmshorn, Germany). "LOGIKA-Baumeisterspiel" is registrated trademark. Size: 25mm (elemetary cube) Material: wood Today the 'Baumeisterspiel' is made by Ingo Uhl GmbH, Logika-Spiele. It is made of recycling-plastic and available in different colours and sizes. Normal size 18mm. http://www.uhlgmbh.de/ The Ingo Uhl company http://www.uhlgmbh.de/baumeis.html The 'Baumeisterspiel' It comes in an 5*5 frame. So many figures have a 5*5 square at the bottom. The pieces are the Soma set plus the I3. Notation of the pieces (height notation): 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 5-square based figures with D4-symmetry (square-symmetry): ---------------------------------------------------------- 1 1 1 1 1 1 1 2 1 1 1 2 2 2 1 1 1 2 1 1 1 1 1 1 1 figure possible (13 solutions) - - - - - - - - - - - - - - - - - - - - - - 1 1 1 1 1 1 2 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 impossible as the 'N' has no legal position. - - - - - - - - - - - - - - - - - - - - - - 1 1 2 1 1 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 2 1 1 impossible Proof: First use the weighting function A (count the cubes at the bottom only): 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 1 0 1 The maximal function values in legal positions are: 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0 1 2 1 1 1 1 1 The sum of the maxima is one smaller than the value of the figure. Therefore the figure is impossible. - - - - - - - - - - - - - - - - - - - - - - 2 1 1 1 2 July, 1995 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 2 impossible No impossiblity proof by weighting functions is possible as the following near-solutions show (only the bottom layer is shown): h h e + f +=ef g g b h h +=ce h e e f a a g b b h c d d d a a d d d - c b b g a a e e f f averaging solutions for c - b g g c c + e f point-sym and \-sym. 5-square based figures with D2-symmetry: ---------------------------------------- 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 impossible Proof: (14. May 1997) First use the weighting function A (count the cubes in the piles): 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 The maximal function values in legal positions are: 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 1 2 2 2 The sum of the maxima is equal the value of the figure. Therefore all pieces must be placed with there maximal A values. Second use the weighting function B: 0 0 1 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 The maximal function values in legal positions with A value maximal are: 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 0 1 0 1 1 1 0 The sum of the maxima is one smaller than the value of the figure. Therefore the figure is impossible. - - - - - - - - - - - - - - - - - - - - - - 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 figure possible (3 solutions) - - - - - - - - - - - - - - - - - - - - - - 1 1 1 1 1 1 1 2 1 1 2 1 2 1 2 1 1 2 1 1 1 1 1 1 1 figure possible (3 solutions) - - - - - - - - - - - - - - - - - - - - - - 1 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 2 1 1 1 1 figure possible (8 solutions) - - - - - - - - - - - - - - - - - - - - - - 1 1 1 1 1 2 1 1 1 2 1 1 2 1 1 2 1 1 1 2 1 1 1 1 1 figure possible (12 solutions) - - - - - - - - - - - - - - - - - - - - - - 1 2 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 impossible No impossiblity proof by weighting functions is possible as the following near-solution shows (only the bottom layer is shown): d d d e e +=ef g g e + c a g f f c a - b h c averaging-solution for symmetries a b b h h point-sym and /-sym. 5-square based figures with C4-symmetry: ---------------------------------------- 1 2 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 2 1 figure possible (6 solutions) 5-square based figures with a single symmetry: ---------------------------------------------- 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 unique 4 sol. 14 sol. impossible 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 10 sol. impossible unique 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 27 sol. 27 sol. 15 sol. 2 1 2 1 2 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 2 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 6 sol. 2 sol. 21 sol. 8 sol. 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 4 sol. 12 sol. 6 sol. 2 2 1 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 2 1 2 1 1 2 1 1 2 1 2 1 1 1 2 unique 7 sol. 36 sol. 3 sol. 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 2 1 1 1 2 3 sol. 15 sol. impossible 7 sol. 34 sol. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 2 2 2 1 1 1 2 2 2 1 2 2 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 124 sol. 54 sol. 13 sol. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 2 1 2 1 2 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 2 1 2 1 1 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 sol. 7 sol. impossible 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 2 2 2 1 1 1 2 1 1 2 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 41 sol. 21 sol. 9 sol. 4 sol. 10 sol. 15 sol. 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 18 sol. 16 sol. 3 sol. 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 2 1 1 1 2 1 2 1 1 1 1 1 1 1 2 1 1 4 sol. 25 sol. 10 sol. 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 2 1 2 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 2 2 1 1 1 1 1 2 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 2 1 2 1 2 2 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 10 sol. 45 sol. 64 sol. 5 sol. 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 2 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 1 1 2 2 2 1 1 2 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 2 1 2 1 1 2 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 2 1 1 53 sol. 28 sol. 29 sol. 16 sol. 14 sol. 14 sol. 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 2 12 sol. 25 sol. 7 sol. 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 1 1 2 1 1 1 1 1 2 1 1 1 1 1 1 2 1 1 1 1 2 impossible 4 sol. unique 9 sol. impossible 3 sol. 1 2 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 6 sol. unique impossible impossible 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 impossible impossible impossible impossible 1 1 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 impossible impossible unique impossible 1 1 3 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 impossible impossible impossible impossible 3 1 1 1 2 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 2 1 1 1 3 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 1 2 1 2 1 1 1 1 1 1 1 1 1 1 1 1 2 1 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 16 sol. impossible impossible impossible -- mailto:Torsten.Sillke@uni-bielefeld.de http://www.mathematik.uni-bielefeld.de/~sillke/