Correct solutions came from John Snyder, USA; Dan Dima, Romania; Philippe Fondanaiche, France; Lou Cairoli, USA; Ron Welch, USA.

This is a very difficult problem.  I've been thinking about posting it for a number of years, and finally got the courage to do it.  The answer is:  There are 312426715251262464 possible legal drawings, about 12.8% of all possible drawings.  A short paper I wrote to explain this is here.