15 puzzle¶
A four-by-four sliding-tile puzzle whose reachable configurations form one parity class under moves of tiles into a single empty cell.
Core Idea¶
The puzzle has fifteen labeled tiles and one blank; only half of all permutations are reachable from a given state, with solvability determined by permutation parity together with blank-row convention. Each legal move swaps the blank with an orthogonally adjacent tile, generating a state graph whose invariant partitions arrangements and whose shortest paths define solution difficulty. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
15 puzzle belongs to recreational mathematics and is useful where the analyst can specify the typed recreational mathematics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the board size, labeled tiles and blank, legal adjacency moves, initial and goal states, parity convention, reachability and any optimal-move metric are explicit. The scope is broad within that domain but bounded by the need for the board size, labeled tiles and blank, legal adjacency moves, initial and goal states, parity convention, reachability and any optimal-move metric are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the board size, labeled tiles and blank, legal adjacency moves, initial and goal states, parity convention, reachability and any optimal-move metric are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name 15 puzzle can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to 15 puzzle. 15 puzzle compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed recreational mathematics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the board size, labeled tiles and blank, legal adjacency moves, initial and goal states, parity convention, reachability and any optimal-move metric are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of recreational mathematics because they reuse the typed recreational mathematics carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Each legal move swaps the blank with an orthogonally adjacent tile, generating a state graph whose invariant partitions arrangements and whose shortest paths define solution difficulty., and type the carrier, state every parameter and convention in the definition, test that the board size, labeled tiles and blank, legal adjacency moves, initial and goal states, parity convention, reachability and any optimal-move metric are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction 15 puzzle Domain-specific
Parents (1) — more general patterns this builds on
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15 puzzle is a kind of State and State Transition Prime
The proposed strict upward parent is
prime:state_and_state_transition.
Hierarchy path (1) — routes to 1 parentless root
- 15 puzzle → State and State Transition → Phase Space
Neighborhood in Abstraction Space¶
15 puzzle sits in a crowded region of the domain-specific corpus (24th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Digit Properties & Recreational Numbers (6 abstractions)
Nearest neighbors
- Mathematical puzzle — 0.92
- Amenable number — 0.92
- Recurrence relation — 0.91
- Number line — 0.91
- Taxicab number — 0.90
Computed from structural-signature embeddings · 2026-09-08