# Message #1506

From: Andrey <andreyastrelin@yahoo.com>

Subject: M12 puzzle

Date: Sun, 06 Mar 2011 16:32:12 -0000

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 congruent pieces like in M24, so search of operations that don’t move some balls on this ring is more tricky.

I took operations: "I", "IMI", "IM2I" and "IM3I". Calcucated for them sequence f(x)=op(x+1)-op(x), where + and - are operations on Z11. If op(x) or op(x+1) is 1 (that is outside of the ring), then f(x) is undefined. If for two functions have congruent segments of values of length k, (that is f(x)=f1(y), f(x+1)=f1(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=IM5IMIM2, C1=IM8IMIM3, C2=IM9IM6IMIM6, C3=IM10IM9IM3IM8. First of 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 puzzle). If we fail this step, we use C1, C2 or C3 to add a ball after 4th ball in sequence (and drop 1st ball), and try C0 again.

Something like that.

Andrey