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Background:So I made a small game played on graph paper and Id like to know what is its if the game can go on forever or if theres a limit to how many moves can be played.Any extra detail or fact about this game is aprecciated

The game itself is pretty simple:Start with a NxN divided in squares.In the first turn a player puts a knight from chess in this board and adds a square anywhere that touches at least one of the already placed squares in the NXN lattice.After this the player moves the knight the same as it moves in chess and eiminate that square (ergo one space horizontally and one space diagonally or reverse)and add another square in the directly besides of the last placed space.The knight cannot visit any previously visited squares.How many moves I make is my score.For example if I am able to make 20 moves my score in that game would be 20. What I am asking for is the maximum number of plays for any nxn board.I am particulary interested in the results for the 3x3 and 4x4 boards. example game:Assume 3x3 board to notate the squares we will use generalized chess notation(a,b,c and 1,2,3)the player places his knight at A-1 and place a new square at the newly created row of D-1.After this the player moves to B-3 and places a new square at D-2.knight to C-1 new square placed at D-2,knight moves to A-2 and a new square is placed at D-3.Knight to C-2 new square at D-4 creating a new column the knight can go to.The knight goes to D-4 and a new square is placed at C-4.Play continues like this until the player has no legal move REMEMBER:the knight cannot go to previously stepped on squares

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  • $\begingroup$ Your description is unclear. What does "add a space" mean? What does "in the clockwise direction" mean? $\endgroup$ – NovaDenizen Oct 22 '16 at 18:17
  • $\begingroup$ @NovaDenizen.I am not quite sure how to explain it,english isnt my first language but ill try my best. $\endgroup$ – Victor Gonzalez Oct 22 '16 at 18:19
  • $\begingroup$ Perhaps you can give a small example (starting with a $3\times3$ board) that illustrates the precise rules for adding squares that you have in mind. $\endgroup$ – Greg Martin Oct 22 '16 at 18:57
  • $\begingroup$ @GregMartin Is that better? $\endgroup$ – Victor Gonzalez Oct 22 '16 at 22:36
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I don't have a complete answer, but I can show that if the starting area is at least 6x6, you can go on indefinitely.

Consider the following two knight's paths. This first one starts in one corner and ends in another.

   1  20   9  14   3  22
  10  15   2  21  30  13
  19   8  31  12  23   4
  16  11  18  27  34  29
   7  26  35  32   5  24
  36  17   6  25  28  33

This one starts one square in from the corner, and ends on another square just inside another corner.

  29  12  15   8  23  10
  14   1  30  11  16   7
  35  28  13  22   9  24
   2  21  34  31   6  17
  27  36  19   4  25  32
  20   3  26  33  18   5

You can start your solitaire game in the corner of a 6x6 square. Follow the first path, all the while adding on squares to fill in the 6x6 region below your starting one. When you finish on the corner of the last path, jump to the starting position of the next path in the 6x6 you have just filled in below. When you finish the second path, you can start over again in the upper left corner of the next 6x6 region. You can continue likewise indefinitely.

You can also rotate and flip the paths so that you follow a spiral to eventually fill in every square in the infinite lattice.

This general process may be possible with smaller shapes, or possibly with nonrectangular regions, I haven't exhaustively tested them.

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A 5x5 region serves too. You just need to use a reflection or rotation of this path, and you can jump into the corner of any neighboring 5x5 region next.

   1  12  23  18   3
  24  17   2  13  22
  11   8  21   4  19
  16  25   6   9  14
   7  10  15  20   5
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