#idea #permutation #braid #3-state #evolution It is possible to resolve a thimble shuffle without tagging the cup of interest. It is possible just by marking whether the two interfaces were triggered by a passage of a cup from one position to another. This proposal works only in the case where the steps of the shuffle are permutations involving only two cups at a time. ![[A Tag-Free Approach to Resolving Thimble Shuffles Leads to Reduction of Chronological Braids 2024-05-17 16.08.59.excalidraw.png|center]] The ball can only be under one of the cups. This is a 3-state system where for convenience we note the states as, $\ket{L}$ , $\ket{M}$ and $\ket{R}$. ![[A Tag-Free Approach to Resolving Thimble Shuffles Leads to Reduction of Chronological Braids 2024-05-17 16.36.24.excalidraw.png|100%|center]] The operations allowed in the proposed situation always keep one cup invariant. And thus, I will use it as a label. As an example, the $R$-operation keeps the right cup untouched and switches the left and mix cup. ![[A Tag-Free Approach to Resolving Thimble Shuffles Leads to Reduction of Chronological Braids 2024-05-17 16.29.44.excalidraw.png|center]] Using the notation above a shuffle can be represented as a chronological braid describing the evolution of 3-cup system. ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-17 23.49.32.excalidraw.png|center]] Exploring the various ways these operations combine I realized few insights about this type of systems. # 2-Operations are Shifts While building multiplication tables for these operations I noticed that their result is state-dependent. The only explanation to their irreducibility is that they are a new kind of operations. ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-18 00.13.02.excalidraw.png|center]] Upon close inspection, 2-operations model 1-D rotations, or rather, shifts. And they are of two type, either right- or left-shifts. By inspective the label of the 2-op one can tell what kind of shift it produces. ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-18 00.21.54.excalidraw.png|center]] Based on the above algebra one can define reduction rules for these shifts. ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-18 00.37.39.excalidraw.png|center]] # 3-Operations Act Like One Performing 3-operations on a state resolves in the same way as if we performed just one operation. ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-18 00.54.17.excalidraw.png|50%|center]] # Operation Reduction is Analog to Untangling the Chronological Braid ![[Resolving Thimble Shuffles via Chronological Braid Reduction 2024-05-18 01.03.24.excalidraw.png|100%|center]]