puzzle puzzle

Description

The configurations of the 8-puzzle game board can be represented as permutations of 9 elements with each element being a digit from zero to eight, taking the empty square as the digit zero. We define the set S9 as the set formed by all the permutations of 9 elements, where each element is a digit from 0 to 8. The number of elements in the set S9 is a factorial of 9:
| S9 | = 9!
In this way, choosing a specific target configuration is equivalent to choosing a certain permutation from the set S9. Let's call the Current Target Configuration –COA from now on - the selected configuration; Considering this selection, it can be shown that the following proposition is fulfilled:
"Only half of the possible configurations on the 8-puzzle game board are capable of reaching the set COA in a finite number of steps."
This proposition, the proof of which I am not going to show, supposes the creation of a bipartition in the set S9.One of the partitions is formed by those permutations that represent configurations from which the fixed CAO can be reached, and the other by which it cannot be reached. they can never reach. Each partition is made up of half the elements of S9:
| S9 partition | = 9! / two
Calling one partition the "blue partition" and the other the "green partition".
The green and blue partitions are in such a way that they have no elements in common; furthermore, the configurations within the same partition are achievable with each other. These two properties imply the fact that any configuration is liable to become COA within and only within the partition to which it belongs. In other words, a configuration set in advance as COA is reachable only by the configurations that belong to the partition in which the selected COA is located.
The game interface consists of the following features:
Game board. Allows token movement in order to reach the current target setting (COA)
Difficulty selection. In the center of the screen, at the bottom, is a bar with a button. By moving the button you can change the difficulty of the game board. The maximum difficulty is 31 steps.
COA board. The COA board serves as a reference so that the player knows where to go. It also allows the change of COA. Pressing on one of the tabs or empty box changes the COA distribution.
Partition change. It is activated by pressing the icon in the game window bar. The partition change allows access to more boards, both startup and COA.
Indicator with needle. The gauge needle always points to the remaining steps to get to the COA from the current dash layout.
Top sign. They are two two numbers separated by the colon sign. The number on the left is the remaining steps to get to the COA from the current layout of the board. The number on the right is the steps that have been carried out since the initial setup of the game board.
Lower sign. It is a number that indicates the number of steps that were missing in the initial configuration of the game board, just before making any movement.
We must know what the 8-puzzle game consists of; We will begin by defining the term "TARGET CONFIGURATION": "The purpose of the 8-puzzle consists of rearranging a certain initial configuration of eight square tiles on a 3 x 3 board, numbered from 1 to 8, in a specific objective configuration". The reorganization of the tiles is carried out by successively sliding the tiles into the orthogonally adjacent empty square. Sliding a chip into the empty box is called "STEP". A key concept related to the target configuration is “PARTITION”.