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IPP 22  Antwerp 2002

Exchange Puzzle of Claus Michael Ringel

Four Problems: TWO and TWO

Design:        Claus Michael Ringel
Puzzle Maker:  Jean Claude Constantin

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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

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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

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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).

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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
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