i'm only scratching the surface with the title, i'm not really
going to state the orthodoxy behind a mathematical matrix,
i.e. e.g.
             [ 2  3  1 ]          uttered 'one by three'...
if i had a two-line bracket i could write it as
               [ 2 3 1
                 1 2 3 ]      uttered 'two by three'...
                 but i'm still fascinated by sudoku, and i can't
get my teeth into it, well **** & proper...
                                to my tally... only one fiendish solution...
but also: sometimes the difficult tier is easier than a mild
tier puzzle.
                      anyway... i just wanted to stress that sudoku,
is an irregular matrix...
         one explanation is: it's a 2 dimensional object,
but it's a 3 dimensional subject,
           in that yes, it's on a piece of paper...
     but as a 3 dimensional subject,
           the concept includes the 2 dimensional object,
but the added dimension, which makes it 3 dimensional
is time... the time it takes to complete such a puzzle.
                    and while you're doing one of these, and getting a buzz
off some **** fine *** (all spice infused) -
    you hit a point where you either (a) become slightly cross-eyed
      or (b) you're looking at the puzzle as if under water and
          it's all blurry
                thus (c) a blind-spot emerges, and suddenly a few
      squares disappear for what could be as much as a second...
               and then you make mistakes...
             plus, if you're doing it at night? all the worse for wear.
so why do i mean a sudoku is an irregular matrix...
  well... i should say "matrix" since i'll include χ (chi) / multiplication
in the notation:         9 x 9 = 81       that's already suspicious
       it's an uneven number, but the puzzle is a square...
  anyway, the matrix:

           [ 9 x 9
             3 x 3
             3 x 3 x 9 ]
                                   nine squares, in each of the nine squares
                  another nine squares,
                                 but then there a need to do the following
to see the optics of the puzzle... i.e.:

                  9 x 9 = 81    +     3 x 3 = 9       +     3 x 3 x 9 = 81
   = 171
                          but then there's the second eye (and the above
      stated whims of doing one drunk):

                                               [ 9 x 9
                                                 3 x 3
                                           9 x 3 x 3 ]

         and as above          171 + 171 =      342...
                  and to my ability to understand the puzzle,
        there are this many variations of inserting a single number into
                     a sukodu - in the fiendish tier...
or at least that was what i was conjuring when i was stuck
   on no. 9019 - and it allowed me to insert a tiny addition (a 3)
   into the puzzle. obviously the number of variations decreases
in the lower tiers.