Knight's Tour Problem

Which is the utility of knight's tour problem for beginners?

None.

To help them better understand how the N moves.

             To show how powerful the knight is.  Pawns can't go to every square. Neither can bishops, or kings.

        He can't put himself in check. A knight CAN sacrifice himself.

I believe the utility may increase, depending on the level one does it at, i.e:

1) Basic , from any square.

2) End where you began.

3) Let some one else or a computer designate beginning and ending squares.

There isn't any. It's just a cute little math model that's been hidden in the game of chess. It certainly wasn't a part of the design of chess 100's of years ago. 

....and it adds to the spectacle of chess (for those who are easily impressed).

I think you are probably right, TGL, as I'm sure there are many people, who  once they know the system, who can do a knight's tour in seconds, but are not that good at chess, and vice versa.

I would like to mention though, that this mathematical problem was first mentioned around 1000 years ago in an Indian text.  

As TGL said, the Knight's tour has more mathematical application than chessic application.

Although one could argue that it may help develop certain similar areas of the brain, i.e pattern recognition etc.

Kind of like cross-training for the brain.