M12 is more difficult.
First, we see that Merge operation has order 11, so we can arrange balls =
in pattern similar to M24: ring (12,2,3,5,9,8,10,6,11,4,7) and the separate=
ball 1. The problem is that operation "Invert" on this ring has no congrue=
nt pieces like in M24, so search of operations that don't move some balls o=
n this ring is more tricky.
I took operations: "I", "IMI", "IM2I" and "IM3I". Calcucated for them seq=
uence f(x)=3Dop(x+1)-op(x), where + and - are operations on Z11. If op(x) o=
r op(x+1) is 1 (that is outside of the ring), then f(x) is undefined. If fo=
r two functions have congruent segments of values of length k, (that is f(x=
)=3Df1(y), f(x+1)=3Df1(y+1), ...) than we can get combination like op1*M^p*=
op2^(-1)*M^q that has k+1 fixed points.
By this method I got operations
C0=3DIM5IMIM2, C1=3DIM8IMIM3, C2=3DIM9IM6IMIM6, C3=3DIM10IM9IM3IM8. First o=
f them has fixed points 2,3,5,9, and others have 3 fixed points. With such =
operations I easily got 4 balls that go in right sequence by the ring. With=
50% probabilty we can add 5th ball to this sequence by C0 (and solve the p=
uzzle). If we fail this step, we use C1, C2 or C3 to add a ball after 4th b=
all in sequence (and drop 1st ball), and try C0 again.
Something like that.
Andrey=20