Reproducible branch analysis
Dead-End Recovery Case Study: Undo, Branch, and Shuffle
An initially solvable board can still reach zero moves. On Turtle seed 20260614, the legal sequence below removes 34 pairs and stops with 76 tiles present but no matching pair among the free tiles.
The useful lesson is not that one face is always dangerous. The evidence shows where options contracted, how far Undo must travel before a genuine alternative appears, and why Shuffle is a different operation from returning to the generated path.
The tested 34-pair branch
1 116-143 2 117-127 3 0-72 4 71-142 5 1-87 6 104-138 7 122-130 8 11-103 9 10-137 10 86-131 11 9-12 12 84-110 13 63-85 14 30-99 15 54-136 16 20-105 17 115-141 18 123-135 19 29-41 20 31-62 21 28-83 22 64-124 23 98-106 24 8-132 25 73-133 26 27-139 27 82-140 28 126-129 29 70-97 30 53-134 31 32-125 32 92-102 33 96-114 34 52-93 => 76 remain, 0 legal pairs
A legal move is not the same as a safe branch. The sequence never removes a blocked tile or mismatched face. Its failure arises from the relationship between the faces left behind and the positions that remain accessible.
Replay Same Board is the cleanest way to study this branch because it reconstructs the exact initial assignment. A random New Game is not equivalent evidence.
Where the option count collapses
The last seven transitions show a narrowing funnel. Counts are recomputed before and after each listed pair.
| Step | Pair and face | Legal pairs before | Legal pairs after | Tiles remaining |
|---|---|---|---|---|
| 28 | 126–129 · 9 of bamboo | 5 | 6 | 88 |
| 29 | 70–97 · 8 of circles | 6 | 5 | 86 |
| 30 | 53–134 · 8 of bamboo | 5 | 5 | 84 |
| 31 | 32–125 · 5 of bamboo | 5 | 3 | 82 |
| 32 | 92–102 · 6 of circles | 3 | 2 | 80 |
| 33 | 96–114 · 8 of characters | 2 | 1 | 78 |
| 34 | 52–93 · 5 of characters | 1 | 0 | 76 |
Why one Undo is not enough
Undoing only step 34 restores pair 52–93, but that pair is again the only legal move. Replaying it recreates the same zero-option state. Two or three undos expose more pairs, yet the tested recovery point is four undos, which returns to the state immediately after step 30 with 84 tiles present and five legal pairs.
At that point the available pairs are 32–61, 32–125, 52–93, 61–125, and 92–102. The failed branch chose 32–125. Choosing 32–61 instead changes which copy of 5 of bamboo remains available and gives a tested route to completion.
A verified continuation after the four-move rewind
32-61, 52-93, 51-102, 50-88, 49-111, 13-69, 2-55 3-118, 65-121, 81-120, 4-80, 40-89, 21-39, 42-79 43-92, 119-125, 90-128, 5-68, 66-112, 44-91, 14-19 18-107, 17-26, 6-45, 38-46, 56-96, 25-109, 37-95 94-101, 22-60, 67-114, 15-36, 48-108, 47-74, 75-100 33-59, 16-76, 7-34, 35-113, 23-77, 24-57, 58-78
The completion is evidence that a recoverable branch exists at the four-Undo point. It does not claim this is the only completion, the shortest completion, or a sequence a Hint control will reveal.
The decisive comparison is between pairings of duplicate 5 of bamboo tiles in one particular state. On another seed, the relevant face and coordinates can differ.
What Shuffle can and cannot recover
Shuffle keeps all currently present positions and redistributes their existing faces using a derived seed. It may create an immediately legal pair, so it is a practical way to continue playing when Undo is unavailable or the player does not want to retrace the branch.
Shuffle does not return to the tested recovery line and does not rerun the initial solution-path construction. A current move after Shuffle is therefore not evidence that the whole shuffled state has a completion. For controlled comparison, use Undo or Replay Same Board; use Shuffle as a new arrangement with a narrower guarantee.
A state-based recovery checklist
When the board reports no moves, record the remaining count and inspect the last several move summaries. Undo until the available set contains a real alternative rather than merely the pair you just restored. Compare pairings that use different copies of the same face, then watch whether the next-state legal-pair count stabilizes or continues to contract.
- One Undo can be mechanically valid yet return only the forced move that caused the stop.
- A higher legal-pair count is diagnostic evidence, not a proof of eventual completion.
- Replay preserves the initial assignment; Shuffle changes the remaining assignment.