Vetka (the name is transcription of russian word
“ветка” which means “tree branch”) is an
game invented by GAMOS. It is played on a
square field divided into square cells.
In the course of the game fragments can not be moved from their initial
locations but only rotated within their cells. The purpose is to reconstruct
a connected single-piece structure without loops.
In a customizable version of the game you can choose field
size, different tileset and also allow generation of the structures with
a “+”-shaped tile.
Uniqueness of the solution
Is the solution unique? My first guess was “of course not”. I thought there are
many ways to rotate tiles to end up with different trees. For example swastika
shaped tree (3x3) allows 2 different assemblies. But after playing several
games I found out that all of them had unique solution, especially those that
did not have “+” shaped tile in it. I tried to build 2 different trees which
differ only by rotations of some tiles and does not contain “+” shaped tiles,
So, my second guess was that such trees can be reconstructed in a
unique way. Then I tried to prove this and was trying to until one day computer
generated a puzzle for me that allowed 2 different solutions. 8 tiles in upper
right corner can be rotated to produce different tree. This tree can be reduced
to 4x4 tree, which leaves me with a question: what is the smallest possible
tree (with no “+” tiles) that allows 2 solutions? Is there such 3x3 tree? 3x4?
I programmed this puzzle during the period when I believed my second guess is
correct. As a result I detect end of game in the following way: I generate tree
and then I keep count on the number of turns for each tile. (Scrambling is done
by performing random number of turns on each tile.) I assume that tree is
reconsructed when each tile returns to its original position. This leads to
unfortunate consequences that one day you will build a tree and this will not
be detected by the program. This means that you were extremely lucky and found a
tree which allows at least two solutions and you've just found “another” one.
On its highest difficulty level original game from GAMOS can be played on a
toroidal field (I haven't implemented this variant). It definitely adds to
difficulty. Original variant also does not fill the entire game field with
tiles. It is hard to say, if this makes puzzle more difficult or not, but it
produces more visually appealing image of a completed puzzle.
Another obvious variant is to use hexagonal tiles instead of square ones. This
one I have implemented. I call it Vetka6X. It is
more difficult than original Vetka. I do
not know if there are any commercial implementetions of this puzzle.
There is one more tileset that covers the plane -- triangles. My original
estimate is that since there are only 4 types of tiles and this includes empty
and Y-shaped tile (both of which are not affected by rotations), this variant
will not be very challenging. But I will definitely try to implement it one day
and check this hypothesis.
Plane can be tiled by regular polygons only in three different ways, so, no
more variants of this type are possible. However, if we move from regular
euclidean plane to non-euclidean plane, we arrive to infinity of possibilities.
Non-euclidean plane allows infinitely many tilings by regular polygons, giving
infinitely many variants of Vetka puzzle.
Creating user interface for displaying and manipulating figures in
non-euclidean plane is a challenge. It is not impossible and there are
implemetations with usefull applications (lookup “Hyperbolic Tree” with your
favorite search engine for an example).