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[alanr555]

This puzzle presents little difficulty if the Mandatory Pairs approach
is applied as a preliminary to writing the profiles.

M/P resolves
1 and 7 on row one
7, 5, 1 on row three
7 and 6 on row four
6, 9, 7 on row seven
1, 7, 6 on row nine

Working on the profiles leads to
4 in rows two , three, four, five and six
9 in rows five and six

This leaves the 3 exposed in row one
and the presence of the M/P marks makes
the resolution almost a "write-in".

In order not to "spoil" the solution, I have not
included the cell details but the above should
enable anyone wishing to use M/Pairs to gauge
progress on applying the methodology.

As additional hints, the following pairs can be
identified in the nine regions - numbering them
across first.

ONE - 3 and 6
TWO - 6
THREE - 1,4,7,8 (and 2 later)
FOUR - 3 (and 9 later)
FIVE - 1,2,4,8,9
SIX - 4 and 6
SEVEN - 1,3,5,6,8,9
EIGHT - 1,4,5,7
NINE - 3,6,7,9 (and 5 later)

Incidentally, regions five and seven are examples of
"definitionally complete" regions in that ALL of the
digits are constrained each to two cells.

In such cases the M/Pair details become the SAME as
the traditional "pencil mark" profiles - and so may be
copied from bottom left to top left when the PMs are
being derived - another saving on the derivations!

Rows seven, eight and nine have examples of the
"mutual reception" where two digits are each constrained
to the same two cells. This is a powerful phenomenon
as all other digits are then excluded from those cells. Of
course that rule applies also with the pencil mark patterns
but is very much clearer with the M/Pairs. As an example:
three cells with pairs marked as (123)(23)(1) immediately
develop to a definite 1 in the third cell and a mutual reception
of 2 and 3 in each of the first two. The exclusion of the 1
from the first cell DIRECTLY resolves the third cell. Also,
once the mutual reception is identified, it "blocks" the use
of those two cells for any other digit - and so can limit the
scope for placing other digits in such a way that only two
cells (or even less than two!) remain possible placements.
The Mutual Reception, thus, can be regarded as being two
resolved cells when counting unresolved cells in the region
(or on the line if the M/R cells are both in line).

This puzzle includes a triple in one region. One of the lacks
of M/Pairs is that it does not highlight triples or pairs that
cross any regional boundary. The only way to deal with them
is to make a note outside the grid. My convention is to mark
the unresolved cells in a line (row or column) as a string of
digits within parenthesis eg (1234567) but to highlight a pair
or triple by using multiple parenthesis eg (36)(12457) for
a pair (either M/Reception or remote) or (247)(1356) in
the case of triple. These notes are not available using the
on-line version but the large print on A4 is perfect - using the
space above the grid and to its right.