Legal moves: identify the blank's neighbours
Start by separating the moves the board permits from the moves you would like to make. The tile and blank exchange places; no other tile changes during a classic slide.
LEARNING CENTRE
Sliding Tiles: plan the blank, not just the next tile
Learn how the blank controls access, how short cycles move tiles around a region, and how to distinguish a difficult position from an impossible arrangement. The short playable studies let you compare first moves against a checked route to the ordered four-by-four goal.
The rules in this version
This version uses a four-by-four grid containing tiles 1–15 and one blank. Only a tile directly above, below, left or right of the blank can slide into it. The goal is 1–15 in reading order with the blank at R4C4. There are no diagonal slides, edge wrapping or whole-row slides. The game creates its scrambles through legal moves from the goal.
Four decisions to understand
Blank routing: prepare the destination first
To move a wanted tile in a particular direction, put the blank on that side of it. This often requires moving other tiles before touching the tile you are trying to place.
Small cycles: return the blank while moving the tiles
A loop can restore the blank to its starting square while changing the nearby tile order. Trace the contents as well as the blank so you know what the loop actually accomplishes.
Reachability: test an arrangement before searching
Some four-by-four arrangements cannot be reached from the goal through legal slides. A counting invariant can identify them without trying every possible move sequence.
Two practice collections
Foundations: read the board before moving
Four studies in legal moves and their immediate effects. Make a prediction, test it, and compare the starting-position answer.
Decisions: compare what happens next
Four studies in future consequences. Distinguish a verified finish, a short tactical proof and a useful feature that does not establish a win.
What the evidence establishes
A legal move need not improve a position. Local routes and cycles demonstrate their stated effects; they do not prove a globally shortest solution. A study's minimum-distance claim requires a search that has ruled out every shorter route from that exact position. It is not a promise about general scrambles. The parity test below applies to this four-by-four board and this particular goal. A route that preserves a solved region is useful only while enough space remains to execute it. Generated scrambles are reachable, but an arbitrary rearrangement supplied in a study may not be.
Each technique has two playable starting positions. The collection and print links reuse those same studies, so a lesson, a practice page and a printed sheet do not count as three different puzzles.
DECISION LAB
Two moves improve the score. Only one stays on the shortest route.
Compare exact distances with a useful but incomplete tile-distance score.
Test your prediction and follow the evidence →