Why there are 576 arrangements
The first category anchors four records. The second category can be arranged among them in 24 ways, and the third can independently be arranged in 24 ways. Multiplying these counts gives 576 complete one-to-one arrangements. The solver can inspect this entire bounded space rather than sampling it or asking an external model for an answer.
Each printed clue is represented as a precise relation: same record, different records, earlier, immediately after or neighboring positions. A case is accepted only when exactly one arrangement satisfies all clues. The shipped case set is also checked with an independently written solver during release verification. That check establishes uniqueness for these cases, not a universal promise about arbitrary puzzles.
What a supporting explanation contains
When you request a deduction, the game considers an undecided relation. It looks for a result shared by every arrangement consistent with the case and your current marks. It then removes premises that are not needed to establish that result. The explanation lists the remaining C-clues and M-marks and reports how many arrangements satisfy them.
Every one of those remaining arrangements must imply the proposed relation, and none may support its opposite. This is an exhaustive constraint proof. The remaining premise set is inclusion-minimal: removing any one listed premise breaks that particular proof. It is not necessarily the smallest possible set or the most elegant explanation a person could discover.
Contradictions and honest reasoning records
If your marks conflict with the clues, the game does not build a hint on a false assumption. Instead it returns premises that have no common satisfying arrangement. The case clues are already known to be consistent, so review the marked relations in that explanation and undo or clear a mistaken deduction.
Applied hints record their supporting premise IDs in the action history. Manual marks remain labeled as manual; the game does not invent a reason you did not provide. Undo is also recorded. The downloadable completed reasoning record preserves these recent actions and final marks. It is useful for reviewing your work, while the spoiler-free share text intentionally omits answers and board state.