The numbers 1 and 18 must be in the isolated squares at the upper right and lower left because we cannot reverse direction and leave a square the way we came. A little experimentation indicates that snaking our way through the grid won't cover all 18 squares. The path we take must cross itself somewhere, moving over a previously numbered square.

I tried working both forward and backward as shown here.

Now try various routes connecting the start and finish. Remember that the path must cross itself somewhere.