In its structure
there is little difference between a tough sudoku and a diabolical

puzzle. The
difference is that there are more places where clues can run out and

more dead
ends.

Take the example
illustrated

here: you can
see the squares

our solver has
managed to

complete using
the strategies we

have discussed
previously. For

clarity’s sake
we’ll ignore all the

‘pencil’ marks
on the grid except

for the first
pair: at 1,5 we have

either a 6 or a
9. This would be

the equivalent
to point *a1*, *b1*or *c1*

on our tough
sudoku illustration.

There is at
least one other pair

our solver could
have chosen on

the grid, but
this was the first, so

let’s be logical
and use that.

Our solver
chooses to try the 6

first, and the
following diagram

shows the
numbers she is able to

complete using
this number.

But with just
two squares to fill,

look at what we
have: at 4,8 the

box needs a 4 to
complete it, but

there is already
a 4 in that row

at 4,6.
Similarly, at 5,4 that box

needs a 6, but
one already exists

in that row at
5,7. No second

guess was needed
to prove that

at 1,5 the 6 was
incorrect.

So our solver
returns to 1,5 and

tries the 9. Now
we are able to

prove the 9 at
2,1, but nothing

else is obvious;
every square is

left with
options. In this case we

could leave both
9s, because we

proved without
doubt that the

6 at 1,5 could
not be correct,

but if the 6 had
simply left us

without
sufficient clues, as the

9 did, we
wouldn’t know which

was true. So,
rather than start a

new, uncertain
path it is better to

return to the
situation we were in

before we chose
at 1,5 and find

another square
to try from. This

is a base we
know to be true.

In this
illustration we can see that

our solver next
looks at square

1,7 where the
choice is between

a 3 and a 6.
Choosing the 3 she

finds herself on
a path that takes

her to just two
more to go . . .

Whoops. We need
a 2 to

complete box 7,
but there’s

already a 2 in
that row at 9,5. In

box 9 we need an
8, but there’s

an 8 in row 8
already. It has

happened
again!