The core mathematical process of the system is solvability determination, which specifies in advance whether or not a given board state can reach the target state for 15 puzzles consisting of 15 tiles arranged in a 4-by-4 grid and 1 blank square.

When a tile array is viewed as a one-dimensional permutation, an inversion calculation is performed that counts the total number of elements in which each tile is located after its original order.

The mathematical reachability of the board is rigorously proven by comparing the parity, which indicates whether the sum of the inverted numbers is even or odd, with the parity of the Manhattan distance that the blank square occupies from the target position to the current position.

The random shuffle algorithm does not simply generate an array using random numbers, but by applying this parity judgment condition backwards, it extracts and generates only the initial arrangement that can definitely be cleared.

This completely eliminates the situation where the user is faced with a layout that cannot be solved theoretically, and always provides a logically complete initial board.

An iterative deepening A-star search algorithm is adopted to derive the shortest steps solution for the generated board. This search method repeatedly executes depth-first search while minimizing memory consumption, and guarantees an optimal solution by gradually raising the estimated cost threshold calculated by a heuristic function.

The compound heuristic function used here is based on the Manhattan distance, which calculates the minimum number of moves required for each tile to reach its target location.

Furthermore, by combining linear conflict calculations that evaluate the additional cost of two tiles whose target positions are placed in reverse order passing each other within the same row or column, the estimation accuracy is dramatically improved.

This pruning process based on the sum of Manhattan distance and linear conflict makes it possible to extremely efficiently eliminate useless search nodes from a huge state space and quickly identify the shortest solution trajectory.

The list of steps derived by the shortest solution search solver is converted into a set of coordinate transformation parameters to visually represent the movement of tiles on the screen.

The board panel movement animation is controlled by an event-driven rendering loop that perfectly synchronizes logical array coordinate updates and physical pixel coordinate transitions.

When a particular tile is swiped or clicked, the vectorial movement component of the adjacent blank square is calculated and a smooth position update is drawn according to the specified easing function.

In addition, the frame-by-frame display function, which allows users to check the calculated shortest step solution in detail, activates a state machine that pauses the continuous playback process along the time axis and separates the state transition for each step as an individual animation frame.

This allows users to visually and step-by-step track the algorithm's selection criteria for their next move.

Since the state space of the 15 puzzle reaches an astronomical scale, efficient use of computational resources in the solution search process becomes an unavoidable challenge.

Our system has an architecture that performs instant calculations in the user's browser local environment, without offloading the entire process of the search algorithm to the server side.

By utilizing runtime compiler optimization of the JavaScript engine and expanding the board state in memory in a one-dimensional array or bitboard format, we minimize the frequency of garbage collection.

Additionally, by implementing a high-speed lookup mechanism for visited states within the browser's main thread by hashing each state node, we have established a zero-latency environment from input to solution presentation that is completely independent of external network delays.

This local calculation-specific design allows users to request complex panel analysis while maintaining a seamless experience.

In order to quantitatively evaluate the user's play process, a strict move count module and an asynchronous timer measurement module linked to state transition triggers are included.

The move counter is designed to increment only at the moment when the coordinate exchange process between the blank square and the tile is logically established, eliminating errors caused by invalid operation input.

Timer measurement records the elapsed time in milliseconds from the first operation to start the game until a perfect match of the target state is detected.

Furthermore, the auto-release animation display function, which helps users who are stuck, stores the optimal sequence of steps cached by IDA star search in a queue structure, and forcibly fires tile movement events by popping them in order at regular time intervals.

While this sequence is in operation, a control mechanism is in operation that temporarily blocks manual interrupt operations from the user and concentrates solely on reproducing the correct route determined computationally.

This system goes beyond simply providing puzzle gameplay and functions as an advanced thinking training environment for comparative verification of human cognitive modeling and mechanical search algorithms.

By comparing the user's own intuitive tile movement strategy with the optimal solution of the algorithm based on heuristic functions, it is possible to consciously train spatial comprehension ability and the construction of multi-step lookahead logic.

In particular, the process of how to bypass local obstacles such as linear conflicts and get closer to the global goal configuration is a living teaching material for the search algorithm learning guide.

By analyzing the frame-by-frame trajectory of the shortest steps solution and self-evaluating the unnecessary loop structure in one's own move selection, students will learn problem-solving methods based on logical reasoning and will form a foundation for deeply understanding the behavior of mathematical algorithms through direct experience.