There were some mistakes with the construction of the odd-ordered magic squares. These mistakes ought to give us hope in finding squares that, if n is odd, you will indeed get (n!)2 magic squares with a random method. I had always maintained that, and until now I never had a reason to question it. And once again, the only reason I questioned it is because there was a mistake in how I remembered the algorithm.
After completing a program written on an Excel 2007 spreadsheet using Visual Basic for Applications, and reading my copy of Reichmann a little more closely, I now see that some corrections need to be made to the articles I have previously written. Corrections are forthcoming, and those articles will be updated.