Brainball

The following images were copied as the site http://www.brainball.de is not necessarily available.

The toy consists of a code wheel, that can be rotated as whole around the sphere, as well as paired sectors can be flipped.

definitions

Be a shift clockwise regarded as positive, and a shift counter-clockwise be regarded
as negative. Without restricting any solutions, be the shifts within -6 and 7.
A rotation is a half turn on the axis, while holding the center piece. Be the
the axis horizontal and the movable part with 4 numbers to the left. Counting the
positions starts on the center piece top left, clockwise.
Be an operation defined as a shift followed by a rotation.
Be a sequence a number of operations followed by a shift.
Be a setting of numbers a vector, the startup vector is 1,2,3,4,..13, yellow.

Points of interest

The space of possibilities is said to be 13! * 2^13 = 51 10^12. Opon some quick trials,
I found several cycles, paths giving the identity. I also found several paths leading to
the same combination.
Hence the questions :

from a given vector, are all possible vectors reachable ?
how long is the longest of the shortest paths ?

Exhaustive search requires some acceleration. Each operation has 13 choices, meaning
with

4 operations, there are 28k choices
6 operations, there are 4.8M choices
8 operations, there are 815M choices
10 operations, there are 138G choices

As newbie I guess a research on the cycles looks promising.

Cycles

Be a cycle the N fold repetition of an operation or the N fold repetition of multiple
operations leading to the start vector.

operation(s)	times for identity
+1/ ( or -1/ symmetry)	8
+2/	12
+3/	4
+4/	12
+5/	6
+6/	14
+7/	14
+1/-1/	20
+2/-2/	12
+1/+2/ ( or +2/+1/ commute)	13
+1/+2/+3 or any permutation	12
+2/+2/ as expected	6
+3/+3/ as expected	2
+3/+3/+3/	4
+3/+3/+3/+3/ as expected	1
+1/+2/+3/+4/	16
+1/+3/+2/+4/	40
+2/-2/+1/	72

Due to symmetry, changing the sign of the shift(s) gives the same result.
There is a great number of short cycles. To fill the space of paths, they have
to be avoided.
It appears strange that unsystematic tests with up to 7 operation to be repeated
showed rather short cycles.

short paths

It was somewhat apparent that multiple path from one vector lead to the same vector.
There seem to be a greater number than assumed. Some unsystematic tests with the
number of path found depending on the search level (= how many operations)

operations	3	4	5	6	7
+1/	3	3	67	67	1443
-1/	3	3	67	67	1423
+2/	3	3	67	67	1422
-2/	3	3	67	67	1417
+1/-1/	1	5	5	114	NA
-1/+1/	1	5	5	115	115

As can be seen, the symmetry is gone, and some searchlevels add
nothing.

To be continued ...

software

finding a path between start and end vector

recursively do a shift and a rotation, then check whether any shift
would match the end vector. Let the shifts run from -6 to +7 and
go to a predefined depth.
I may publish the software at a later stage on one of my delphi pages

simulations
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last updated subpage 27.dec.00

Copyright (99,2000) Ing.Büro R.Tschaggelar