Both tile 4 and tile 8 reduce the Manhattan total from 6 to 5. Are they equally good first moves?
Make a prediction before opening the reasoning.
Compare the predictions
- Yes: both have the same remaining distance.
Not supported. They share a lower bound of 5, but their exact remaining distances are 5 and 7.
- No: tile 4 begins the shorter finish.
Supported. Tile 4 permits a six-move total finish; tile 8 requires at least eight moves in total.
- No: tile 8 is better because it reaches its goal.
Not supported. Placing tile 8 immediately costs access to the upper-right rearrangement.
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The position and the question
The Manhattan total adds each numbered tile’s horizontal and vertical distance from its goal square. It ignores the blank. Since one slide moves one numbered tile one square, this total is a lower bound on the number of remaining moves.
In this reachable position the total is six. Tile 4 at R2C3 can move right, and tile 8 at R3C4 can move up. Each appears to improve the same score by one. We will compare what space those moves leave for the next part of the route.
Show the reasoning
Follow tile 4’s branch
Slide tiles 4, 7, 3, 4, 8 and 12, in that order. Their source squares are R2C3, R1C3, R1C4, R2C4, R3C4 and R4C4. This six-move route reaches the goal. The lower bound was six, so the route is shortest without needing a broader search.
Tile 4 moved right: five moves remain on a shortest finish. Try tile 8 first
Sliding tile 8 up places that tile in its goal square immediately, but leaves the blank at R3C4. The upper-right group containing 7, 3 and 4 still needs to be rearranged. An exhaustive search from the goal finds that this new position is seven moves away. Including the initial slide, this choice needs at least eight moves in total.
Tile 8 moved up: the same Manhattan score of 5, but seven moves remain. Compare the quantities honestly
After tile 4: Manhattan total 5, exact remaining distance 5. After tile 8: Manhattan total 5, exact remaining distance 7. The same lower bound can belong to positions with different true distances. The score has not become wrong; it simply does not encode the interactions caused by routing the blank.
Keep the useful part of the heuristic
The score is still valuable for ruling out a shorter finish: a position with total 5 cannot be solved in four moves. What fails is the extra assumption that every score-reducing move must begin a shortest route. When choices tie, examine the next required blank positions or use a bounded exact search.
The tempting shortcut
Any move that reduces Manhattan distance is equally efficient.
Where the shortcut breaks
Tile 8 reduces the score but leaves a seven-move remainder, while tile 4 leaves a five-move remainder.
A better decision
Use Manhattan distance as a lower bound; compare actual routes before treating a tied score as a tied decision.
Changed conditions
What about sliding tile 3 down from R1C4 as the first move?
Check the changed-condition answer
Its Manhattan total becomes 7, and the exact remaining distance is 7. That branch also requires at least eight moves including the first move. The tile-4 route is the only shortest first choice on this board.
Take this back to your next game
A useful evaluation number can omit the interaction that decides between two moves.
How the claim was checked
Breadth-first enumeration from the exact goal through distance 13, checking all neighbours of this distance-6 state.
- 31,044 unique states visited through distance 13
- Start distance 6 and Manhattan total 6
- After tile 4: exact 5; after tile 8: exact 7; after tile 3: exact 7
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction