You're not going to believe this!  In a frenzy of frustration I attacked the keyboard and madly and randomly pressed key after key - probably 200 keys.  Then I tried the Draw Brush and it worked.  I tried all the other sculpt brushes, and they all worked exactly the way they are supposed to.

The cube responded to each brush stroke.  I quickly took a screen shot of the whole Blender window.

I continued to experiemnt with the sculpt brushes for a while.

Then I closed down Blender and opened it up and tried sculpting another cube.  Again only the Grab Brush worked.  I opened up the saved screen shot to compare it to the Blender window I was trying to work in, and both were identical.

So, now I know that Sculpt Mode is fully operational on my system.  The problem is that I have no idea how I got it to work.  I don't know what keys I hit, or in what combination.  I don't know how to get it to work again!!!!!!