user26877: It is well known that a lone knight can move all around the board covering each square once and only once. Is this possible when the lone knight starts only on certain squares ? How many unique solutions   does this problem have now that powerful engines exist?

user4418: 
Is this possible when the lone knight starts only on certain squares ?

If you stop and think about this for a second you will realize that the answer is obviously "Yes". If you have a "Knight's Tour" (that's what it is called) starting on square X then by definition it also goes through square Y (for all squares Y on the chessboard). Hence the exact same moves, just starting on Y, will also be a Knight's Tour.

How many unique solutions does this problem have

According to Wikipedia:

On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse.


26877: <p>It is well known that a lone knight can move all around the board covering each square once and only once. Is this possible when the lone knight starts only on certain squares ? How many unique solutions   does this problem have now that powerful engines exist?</p>
 4418: <blockquote>
<p>Is this possible when the lone knight starts only on certain squares ?</p>
</blockquote>
<p>If you stop and think about this for a second you will realize that the answer is obviously &quot;Yes&quot;. If you have a &quot;<a href="https://en.wikipedia.org/wiki/Knight%27s_tour" rel="nofollow noreferrer">Knight's Tour</a>&quot; (that's what it is called) starting on square X then by definition it also goes through square Y (for all squares Y on the chessboard). Hence the exact same moves, just starting on Y, will also be a Knight's Tour.</p>
<blockquote>
<p>How many unique solutions does this problem have</p>
</blockquote>
<p>According to <a href="https://en.wikipedia.org/wiki/Knight%27s_tour" rel="nofollow noreferrer">Wikipedia</a>:</p>
<blockquote>
<p>On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections). The number of undirected closed tours is half this number, since every tour can be traced in reverse.</p>
</blockquote>
