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How many solutions does the Knight’s Tour puzzle have when the knight first moves from square a1, through all squares (on each square once), and return back to a1? Please give some solutions. Also, does the Knight’s Tour have any solutions for starting from any random square?

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Wikipedia quotes several sources for a count of 26,534,728,821,064 for the number of closed directed tours of the 8x8 board. As Brian Towers notes in his answer, that is equivalent to the count of Knight's tours starting from some given square (such as a1) and ending on the same square. The same Wikipedia page exhibits several such tours.

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Please share some solutions.

If you use the search function (top left corner of the screen) you will find this answer with a beautiful picture of the moves.

And also has this puzzle solutions from any random square?

If you stop and think about it for a moment you will realise that if the knight starts on a1, moves through all the other squares exactly once before returning to a1 then that sequence of moves constitutes an answer for every square on the board. Hence every solution where the knight returns to the same square is a solution for every square of the board. This was also pointed out by Aric in his answer above.

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