72 out of the 362880 permutations of the nine digits satisfy the criteria for a probability of 1/5040 ~= 0.000198412698412698.

Filtering out rotations and reflections by requiring the top-left number to be the smallest corner and the top-right number to be smaller than the bottom-left number, the following are the nine basic ways the criteria can be met:

DefDbl A-Z

Dim crlf$, dig(9)

Private Sub Form_Load()

 Form1.Visible = True

 

 Text1.Text = ""

 crlf = Chr$(13) + Chr$(10)

 d$ = "123456789": h$ = d$

 Do

   DoEvents

   For i = 1 To Len(d)

     dig(i) = Val(Mid(d, i, 1))

   Next

   good = 1

   For maj = 0 To 6 Step 3

     tot = 0

     For mnr = 1 To 3

       tot = tot + dig(maj + mnr)

     Next

     If tot Mod 9 <> 0 Then good = 0: Exit For

   Next

   If good Then

     For maj = 1 To 3

       tot = 0

       For mnr = 0 To 6 Step 3

         tot = tot + dig(maj + mnr)

       Next

       If tot Mod 9 <> 0 Then good = 0: Exit For

     Next

     If good Then

       tot1 = dig(1) + dig(5) + dig(9)

       tot2 = dig(3) + dig(5) + dig(7)

       If tot1 Mod 9 = 0 And tot2 Mod 9 = 0 Then

         If dig(1) < dig(3) And dig(3) < dig(7) And dig(1) < dig(9) Then

          Text1.Text = Text1.Text & Mid(d, 1, 3) & crlf

          Text1.Text = Text1.Text & Mid(d, 4, 3) & crlf

          Text1.Text = Text1.Text & Mid(d, 7, 3) & crlf

          Text1.Text = Text1.Text & crlf

         End If

         ct = ct + 1

       End If

     End If

   End If

   

   overct = overct + 1

   

   permute d

 Loop Until d = h

 

 Text1.Text = Text1.Text & ct & Str(overct) & Str(ct / overct) & crlf & " done"

  

End Sub