6
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On a $3\times3$ grid we have:

33grid

with $8$ moves needed to swap the red and blue knights.

What is the minimum numbers of moves to swap the knights on a $4\times4$ grid?

44grid

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4
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It will take:

20 moves to swap the knights

This can be done as follows:

Zerris

All credit goes to @klm123 for demonstrating a good way to visualize this in a similar puzzle.

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  • 1
    $\begingroup$ You seem to be missing some connections - B2 to C4 and D3, and C3 to A2 and B1. This does actually put opposite-coloured squares in reach of B2 and C3. $\endgroup$ – Zandar Mar 5 '16 at 7:39
  • $\begingroup$ Right you are - no idea how I missed that. As a result, we can make the solution more efficient. $\endgroup$ – Zerris Mar 5 '16 at 8:05
0
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I believe it is, in chess terms,

20 half moves, or 10 move pairs.

In algebraic notation:

 1.  N3b2 N2c3
 2.  Na1  Nc1 
 3.  Nb3  Nbd3 
 4.  Ncb4 Nc2 
 5.  N1d2 Nd4 
 6.  Nb1  Nd2 
 7.  Na3  Ncb1 
 8.  Nbc4 Nac3 
 9.  Ndb2 Nc2 
 10. Na4  Nd1 
 

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