oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo IPP 22 Antwerp 2002 Exchange Puzzle of Claus Michael Ringel Four Problems: TWO and TWO Design: Claus Michael Ringel Puzzle Maker: Jean Claude Constantin oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo Puzzle Description: Like a cube snake, but as a closed circuit (26 cubes on an elastic string). Four problems: To form a 3x3x3 cube with a specified empty position. Puzzle Classification (Dalgety-Hordern): FOL-HGCL Description of the circuit (and numbering of the cubes): 3 7 3 ... 2 4 6 8 10 12 14 16 18 20 22 24 26 2 ... 1 5 9 11 13 15 17 19 21 23 25 1 oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo For the first two problems, there are several solutions, for example: THE MISSING CORNER: lower middle upper 5 24 23 6 21 22 7 8 9 4 25 26 19 20 11 16 15 10 3 2 1 18 13 12 17 14 THE WELL: lower middle upper 9 8 7 16 17 20 15 18 19 10 11 6 13 12 21 14 22 3 4 5 2 25 24 1 26 23 oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo For the second two problems (THE ENTRANCE and THE HOLE INSIDE), there do not exist solutions! In order to prove this, just use two colours for the individual cubes of a 3x3x3 arrangement, such that neighbouring cubes have different colours. Let the corners be black, thus we use 14 black cubes and 13 white cubes. Neighbouring cubes on the snake have to have different colours, thus the snake consists of 13 black and 13 white cubes. It follows that the empty position has to be black, thus it has to be a corner of the 3x3x3 cube (=MISSING CORNER) or the center of a side (=WELL), but not the center of an edge (=ENTRANCE) or the central cube (=HOLE INSIDE). oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo Claus Michael Ringel, Dieselstr. 5 B, 33613 Bielefeld. http://www.mathematik.uni-bielefeld.de/~ringel/general.html In particular, see: Denkspiele aus aller Welt (2), http://www.mathematik.uni-bielefeld.de/~ringel/puzzle98.pdf oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooo