1. The Forming of Symmetrical Figures from Tetracubes Baiba Bārzdiņa Riga State Gymnasium No. 1

2. The set of tetracubes ILNT OKSZ

3. Tetracubes are used usually to form the given shapes, e. g.

4. 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.

5. Historical references One of the first articles in which attention has been paid to tetracubes is the article by J. Meeus in 1973.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.

6. 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

7. Admissible layers A BCD EFG

8. 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

9. Filters Elementary filters 4 (3) 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

10. 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

11. Solutions found by the computer programme elaborated by A. Blumbergs

12. Results

13. Some important towers 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 BBBB BADAGGCCBBCGAFBAGAAFAG CABCGGGGG