Methods and search [frap-0005]
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    Once the constraints are known, a solver still needs a reasoned way to choose the next move. A solving method organizes a sequence of smaller goals: at each stage, some information has been fixed and some remains free. The useful mathematical question is which moves accomplish the next goal while preserving what has already been achieved. The stabilizer of the recorded data, as in the general construction MulAction.stabilizer, gives language to that requirement; successive stages suggest a chain of subgroups. Conjugation relocates the effect of a known sequence; commutators can help isolate effects, whose actual support must be checked in each example. Salkinder uses conjugates and commutators in the cube setting [salkinder2021nxnxn, Sections 2 and 3.2].

    Explaining why a method works and judging how well it serves a player are related but different tasks. A complete method needs an argument that its stages cover the remaining cases and terminate at a solved state. Practical use also depends on recognition, memorization, and ease of execution. Computational search raises further questions about time, memory, and solution length.

    Cayley graphs describe move sequences as paths between group elements [pan2021fourier, Section 4, Definition 4]; actions on partial information lead to Schreier graphs. The generic labeled-edge construction is Quiver.SchreierGraph: a move labels an edge from a state to its image. For the left action of a group on itself, this is the Cayley viewpoint. These declarations supply stabilizers and action graphs; they do not establish a puzzle-solving method. Such descriptions become useful when comparing reduction strategies or searching for shorter sequences. Any length comparison must specify the allowed moves and their costs. The goal is to connect a method's observable effect with its mathematical explanation. Scrambling and search functions describe a related use of functions on the move group. The complete layer-first construction is given in a complete layer-first method, while computer solving and efficiency turns these graphs into exact searches, lower bounds and subgroup reductions. This chapter keeps the method next to the graph and stabilizer language needed to analyze it.