Baiba Bārzdiņa Riga State Gymnasium No. 1 The Forming of Symmetrical Figures from Tetracubes The set of tetracubes I L N O K S T Z Tetracubes are used usually to form the given shapes, e. g. Description of Problem • The competition work is dedicated to the complicated problem of combinatorial geometry - to find all polycubes having four planes of symmetry and assemblable from tetracubes. • The aim of my work is to solve this bulky problem for wider class of polycubes, namely for polycubes with bases 3x3. Let us explain that a polycube is a solid figure obtained by combining unit cubes, joined at their faces. If the polycube consists exactly of 4 cubes it is called a tetracube. Historical references • One of the first articles in which attention has been paid to tetracubes is the article by J. Meeus in 1973. • A. Cibulis has popularized tetracubes in Latvia, e. g. in the magazine “Labirints” in 1997-1999. •The problem on symmetrical towers was offered in the international conference “Creativity in Mathematical Education and the Education of Gifted Students”, Riga, 2002. Tower with bases 3 x 3 A tower with bases 3x3 is a polycube having four symmetrical planes and which can be inserted in a box with bases 3x3, but which cannot be inserted in a box with bases less than 3x3 Admissible layers B A E C F D G Plan of the problem solving • To find all combinations of layers with the total volume 32. I found 666 combinations by the computer programme. • Obtaining of permutations and their analysis. • Necessary conditions: - filters (Lemma on filters) - method of invariants (colouring) • Analysis of the remaining combinations by the computer programme elaborated by A. Blumbergs Filters 4 (3) • Elementary filters BBBB (1/1) CAAA (2/2) DAAA (2/2) 5 (17) CCCBA (10/6) CCDBA (30/4) CDDBA (30/0) DDDBA (10/0) ECCAA (16/8) ECDAA (30/6) • More complicated filters Lemma on filters A tower cannot contain the following layers: DD, DF, FD, CD, DC, BG, GB, EG, GE, FG, GF, CF, FC, DE, ED, EF, FE, FFF, EEE, GAG, GCG, GDG, EEG. FF, EE, AG, DG, CG cannot be two last layers of a tower. To prove Lemma several nontrivial methods were used: method of interpretation, Pigeonhole principle, and symmetry Solutions found by the computer programme elaborated by A. Blumbergs Results Some important towers BBBB • • • • CABCGGGGG Only BBBB has 5 planes of symmetry Only AAFAG contains F as the inner layer There is a unique stable tower with height 9 Towers GADAB, GCCBBC, GAFBAG can be assemblable only in one way AAFAG BADAG GCCBBC GAFBAG