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I'm working on 8 number puzzle, where 8 numbers are randomly generated and placed in 9 cell board. One cell at bottom right corner is kept empty where we can swap with adjacent (top-bottom-left-right) cells if any available.

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The puzzle can be solved if all numbers are arranged in sequence starting from top left cell. enter image description here

Now after some trials, I found that some arrangements of (randomly generated) numbers in starting of the game are not solvable.

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I think, after generating random numbers, we can find mathematically that random numbers can be solved after placing on board or not.

Can anybody help to find the solution/algorithm/mathematical formula to determine that numbers can be solvable or not?

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This probably fits in better over at math.stackexchange under game theory: math.stackexchange.com/questions/754827/… –  Hao Ye Aug 10 at 18:12
The mathematics is studied using Permutation Groups. Essentially all boards can be solved to either the correct solution, or the pseudo-solution with the 7 and the 8 swapped. An algorithm to generate valid boards might start by applying 1,000s of swaps randomly to a board that is known to be valid. –  Joshua Shane Liberman Aug 12 at 23:40
Or perhaps even better yet: the puzzling stack at puzzling.stackexchange.com –  SQB Aug 15 at 20:08

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