You do not need to be able to play chess to solve
this problem.
<p/>
The standard move for a knight on a chess board
is 2 steps in one direction and one step in the
other direction. A knight's tour is a sequence of
moves in which the knight visits every square on
the board once and only once and a circuit is a
tour in which the knight returns to the starting
point.
<p/>
Prove that a knight cannot make a tour on a 2 by

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math> board for any value of 
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow><mi>n</mi></mrow></math>.

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How many different tours can you find on a 3 by 4
rectangular board?

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<br />
<br />
<img src="mcprob4/knight_def.gif" alt=
"4x3 table with letters a to l" /> <br />
<br />
<br />
<p>Prove that a knight cannot make a circuit on a 3 by 4
rectangular board.</p>