Lege-Wege: A Domino Puzzle Torsten Sillke, Bielefeld
========================== August 1993
The puzzle consits of all black/white colorings of the sements (a,b,c,d,e,f,g)
such that at each crossing point there is an even number of black segments.
-----------
| f e |
|-a-+-g-+-d-|
| b c |
-----------
There are 2^7 combinations without the even condition.
The even condition at each crossing point reduce the
number of possibilities by a factor two. So there are
2^5 = 32 combinations if you fix the domino. The number
of pointsymmetric configurations is 2^3 = 8, as a=d, b=e, c=f
and the even crossing point condition.
The number of really different tiles (rotations allowed) is
therefor (32 + 8)/2 = 20.
The number of endpoints (a,b,c,d,e,f) colored black is even.
This is a consequence of the fact, that in each graph is the
number of vertices with odd degree even.
# black segments at the boundary: 0 2 4 6
# Dominoes: 1 9 9 1
The symmetry in the table is clear, as a bijection given by the
color exchange (black <-> white).
Table of Dominoes:
------------------
0-path domino 3-path domino
----------- -----------
| : : | | | | |
|...+...+...| |---+---+---|
| : : | | | | |
----------- -----------
1-path dominoes 2-path dominoes
----------- ----------- ----------- -----------
| : : | | : : | | | | | | | | |
|...+...+---| |---+...+...| |---+---+...| |...+---+---|
| : | | | | : | | | : | | : | |
----------- ----------- ----------- -----------
----------- -----------
| | : | | : | |
|...+...+...| |---+---+---|
| | : | | : | |
----------- -----------
----------- ----------- ----------- -----------
| : : | | : : | | | | | | | | |
|...+---+---| |---+---+...| |---+...+...| |...+...+---|
| | : | | : | | | : | | | | : |
----------- ----------- ----------- -----------
----------- -----------
| : : | | | | |
|---+---+---| |...+...+...|
| : : | | | | |
----------- -----------
----------- ----------- ----------- -----------
| : | | | | : | | | : | | : | |
|...+---+...| |...+---+...| |---+...+---| |---+...+---|
| | : | | : | | | : | | | | : |
----------- ----------- ----------- -----------
----------- -----------
| : : | | | | |
|...+---+...| |---+...+---|
| | | | | : : |
----------- -----------
Some Puzzles:
1) Build a 3*6 rectangle with the 9 one-path dominoes.
The black segments should build a single line.
2) Build a 4*4 square with 8 of the 9 one-path dominoes.
The black segments should build a single line.
This gives six problems, as you can try to leave each
of the 9 dominoes out. One problem is not solvable, but
you can manage to connect only two segments to the boundary.
3) Tile k of the 9 one-path dominoes in such a way, that the
black segments form a circuit. It is possible to build circuit
with k=3..8 dominoes. Find the easy proof, that no circuit can be
build with all 9 dominoes. Try to build circuit with different
length and different enclosed area.
Which combinations (# dominoes, length, area) are possible?
One segment is one unit-length. This exercise gives you a
hint, why the 9 domino circuit is impossible.
4) Tile a 4*4 square with 8 of all 20 dominoes.
No black segment should touch the boundary.
Which patterns for the black segments can be build.
There are only 10 different pattern, if the zero-path
tile is not used. 29 are possible, if it can be used too.
If you tile the 4*4 square with fewer than 8 dominoes,
- so you get holes - which pattern can you creat?
5) Tile the 6*6 square with the 18 one-path and two-path dominoes.
The black segments should build a single line.
At which places can the endpoints be placed?
Peter Seroka found the first solution by hand. I have drawn the
line of black segments he found. How many tilings give this line?
6) Tile the 6*6 square with the 18 one-path and two-path dominoes.
No black segment should touch the boundary.
This puzzle is impossible, which confirmed Udo Sprute's
general puzzle solver 'arrange'. Is there a short proof?
7) Tile the 6*6 square with the 18 of the 20 dominoes.
No black segment should touch the boundary.
Which combinations are possible?
8) Build a circuit with all 18 one-path and two-path dominoes.
(plus the zero-path dominoe marked '- -'.)
. x x x x x . . . x . . . . x x x x x . .
x x x x x x x x x x x x x x x x x x x x .
x x x x x x x x x x x x x x x x x x x x x
x x x x x x - x x x x x x x x x x x x x x
x x x x x x - x x x x x x x x x x x x x .
. x x x x x . x x x x x x x x x x x x . .
9) Tile a 5*8 rectangle with all 20 dominoes.
You should build one circuit. There are two solutions.
10) Tile a 4*10 rectangle with all 20 dominoes.
A circuit is not possible but a path can be build.
The path may end at a short edge of the rectangle.
references:
- Wer weiss wohin?,
Ein Spiel der Sammlung 'Grips + Co' von Triangle Team, 1993.
- Anaconda, Heye 1989
<- Wenn Knollennasenmaennchen spielen, spielbox 92:3, 34-36, K-M. Wolf
- Irgendwie, Pyramo, 1990?, Dieter Matthes & Silvia Heinz
<- spielbox Denkspiele, spielbox 92:1, 32
- ... lege Spiele!, K. H. Koch, dumont TB 192, 1987
- Klassifikation der Legespiele, fachdienst spiel 5/92
Redaktion: Deutsches Spiele-Archiv, Ketzerbach 21 1/2, Marburg