A loop changes tile order
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
PRACTICE COLLECTION
Sliding Tiles — 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.
Print these four studies with answers
Use a prediction, a test and a reason
- Read the task before opening the answer. Say where a move lands or what it changes.
- Test it on the playable board. In two-player studies, play both sides to inspect a reply.
- Undo and compare one alternative. Use the answer’s stated search limit when explaining the result.
- Repeat the decision on paper, where the move cannot be checked for you automatically.
These are teaching groups, not a player-tested difficulty ladder. The four board IDs match the technique examples, individual study pages and print pack.
Compare a longer cycle
Watch the displaced tiles as you route the blank. Reach the target, then compare your route with the checked shortest route.
Check the odd invariant
Calculate inversions while ignoring the blank. Add the blank’s row counted from the bottom. Is this position reachable from the target? Then solve it.
Parity survives a longer route
Calculate inversions while ignoring the blank. Add the blank’s row counted from the bottom. Is this position reachable from the target? Then solve it.
Bring the idea to a fresh round
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.
Open Sliding Tiles →