Solving for the second number
The fewer empty squares you have in any
box, row, or column, the better your chances of proving the empty
squares, so look for the most populated rows, columns, and boxes.
For example, concentrate on the middle three rows. You have a 6 in
box 5 and a 6 in box 6, but no 6 in box 4, so that seems like a good
number to focus on. The sixes in rows 4
and 5 means that the 6 of box 4 must be at either 4,1 or 4,3, so we
have pencilled in those options. Revealing the rest of column 1 in
Figure 1-5, we find a 6 already in that column. So the 6 of box 4
can’t be at 4,1 and must be in the only remaining square at 4,3. The
second number of our puzzle has been solved.
Figure 1-5. Searching for sixes to solve
the su doku.
Incidentally, when we revealed the rest
of the grid, did you spot the 6 in column 2? If we hadn’t already
excluded a 6 from column 2 of box 4 with our sixes at row 4 and row
5, this six would have done the job nicely. Having such a surfeit of
riches is rare, but worth pointing out.
Cracking open the first box
Moving on in our exploration of the
given clues; look at the 1 in row 4 and the 1 in row 5 in Figure
1-6. Between them they stop any 1 appearing anywhere in box 4 other
than in the only square available at 6,1. No pencilling required
here, it’s the only place for a 1 to go.
Figure 1-6. The 4 at 4,8 means the only
place for a 4 in box 4 is at 5,2.
Now look at the 4 in row 4. It very
nicely stops another 4 from going in box 4 at row 4. Because we’ve
already solved some numbers in this box, only one possible square is
left for the 4, at 5,2. Box 4 is filling up quite nicely and we have
only the 2 and the 5 left to solve. Either of these numbers could go
into each of 4,2 or 4,3. We don’t have an obvious way of proving the
correct square at this stage from the clues provided, so we’re stuck
for the moment.
Before we move on, pencil in the two
options of 5 and 2 for both squares: At some stage we’ll be able to
prove one of the numbers and can solve the box.
We can draw a very important implication from the two unsolved
squares: Both contain either 5 or 2 as we have proved, but that must
mean that these two squares can be the only place for a 5 or 2, not
only in that box, but also for the remaining unsolved squares in
that row. Only one 5 and one 2 can be in the row and we’ve just
proved where they are.
What we just discovered are a matched
pair of twins. A twin is a number that has been proved to appear in
either of two squares that helps disprove its presence in another
part of the grid. A matched pair can help us solve other problems as
the puzzles get harder. Using your clues
By now, you’ve probably familiarised
yourself with the position of given numbers and can dispense with
blocking off parts of the grid, although it’s a useful tool when
you’re concentrating on a specific part of the grid. The ‘good’
clues eventually start to jump out at you as in Figure 1-7. Here we
have a combination of sixes stopping another 6 from appearing
anywhere but in 9,4 of box 5. The 6 of this box doesn’t really help
us to solve any other numbers, so we move on.
Figure 1-7. Separating out the ‘good’
clues.
We’re getting a good handle on the grid
in Figure 1-8. The eights aren’t helping to solve the 8 of box 6
immediately, but they allow us to note that the 8 could be in 4,9 or
6,9. While the eights don’t solve anything immediately, making these
observations is always helpful for use at a later stage of solving.
Figure 1-8. Getting a good grip on the
grid.
 We
can still make plenty of observations and solutions from the given
clues and those squares that we’ve already solved. For example, look
at the 7 in column 4 and the 7 in column 6. Together with the 7 in
row 1 they solve the 7 in box 2.
You have enough clues to start solving
some numbers for yourself, so here’s the grid as far as we’ve solved
it in Figure 1-9. See how much further you can get using the same
simple logic that we’ve used so far.
Figure 1-9. You’re on your own with the
rest of this puzzle. Good luck!
Taking Su Doku Up a Notch
It’s time to start being methodical in
our solving and we have to get serious about discovering the secrets
of each individual square. Depending on the grade of difficulty of
the puzzle you have to decide whether to simply bite the bullet and
write in all the options for every square right now, or take it
gradually box-by-box (or row-by-row or column-by-column). As this
puzzle is only of moderate difficulty, we’re going to take this
discovery process gradually. All the
options for the squares of box 6 have been noted in Figure 1-10. We
found them by asking at each square ‘Will such-and-such number go
here?’ from 1 through to 9 while checking to see if that number is
already in its box, row, or column. Try doing the exercise to check
the numbers for yourself (we’re not perfect, you know!).
We’ve already proved that at 4,7 and 4,9
neither 2 or 5 can appear because of the pair of twins in box 4 that
have already fixed their position.
Figure 1-10. Taking a stab at box six.
Singling out lone numbers
Look at the bottom left hand square of
box 6 and you see that the only number that can go into that square
is a 2. You don’t have any clues around in the rows or columns that
might indicate that a 2 is the solution to that square, and only by
eliminating all the other options could we work out that the 2 goes
in 6,7. We call this discovery of a number by a process of
elimination a lone number. The second
result of solving that 2 in 6,7 is that all other optional twos in
that box, row, and column may now be eliminated. The new situation
is illustrated in Figure 1-11. Having removed all the optional twos,
to the right of the solved 2 you can see that the 5 is on its own @@nd
another lone number solved. Solving the 5 (so all the optional fives
are removed) leaves the 8 on its own in 6,9 and solved, and then go
the optional eights, leaving a lone 7 at 4,9 and so on. Carry on
this process as far as you can with this square. Think about the 3
and the 9 at 5,7. What can you conclude, although there appear to be
two options in that square?
Figure 1-11. Solving for lone numbers.
When you’ve finished your number-fest in
that square, check the consequences of solving those numbers in the
associated rows, columns, and boxes. We hand the puzzle in Figure
1-12 over to you for solving as far as you can, using the techniques
and strategies you’ve picked up so far.
Figure 1-12. Try your hand at this
puzzle, keeping an eye out for lone numbers.
Serious Su Doku Solving
Figure 1-13 demonstrates not only a
number of su doku principles discussed elsewhere in this part, but
also to look beyond the obvious. Consider the number 4:
Figure 1-13. Look beyond the obvious.
-
The fours in columns 1 and 3 and the
4 in square 9,8 preclude a 4 going anywhere in box 7 except 7,2
or 8,2.
-
Because row 8 is full in box 9, the
4 in box 9 will have to go in row 7 @@nd so the 4 in box 7 has
to be in row 8, at 8,2.
-
The 4 in square 5,8 precludes a 4
from going anywhere else in column 8. This means the 4 in box 9
cannot go at 7,8, and since the 4 at 9,5 precludes the 4 from
going anywhere in that row in box 9, leaving only 7,9.
You should see another 4 to solve very
easily in the middle three columns.
You will always have another set of
three unresolved numbers, due to the symmetry of su doku. The
unresolved numbers are always worth checking to see if they can help
to force the resolution of a number. Two
unresolved numbers in a square serve just as well as a solved number
if we know that the number can only go in one of these squares. In
su doku jargon, these lines of unresolved numbers are called twins
and triplets. Extraneous options
Let’s look at some solving strategies
for the more difficult grades of su doku. For these schemes to work
you must have meticulously discovered all the options for every
unsolved square in the grid. We discussed this earlier, when we
looked at the options of a single box, and the method is the same:
Look at each square and ask the question ‘Can such-and-such number
go in this square?’ for each number 1 to 9.
As shown in Figure 1-14, many of the
squares have already been solved, but some remain with their
pencilled options. Every option seems to have an alternative square
that it might go in. So how do we move on? With logic, of course.
Figure 1-14. Looking at the options -
narrowly.
Using a mask helps to concentrate on
column 4. If we look at box 8 there are two sets of options, but we
need to look at the pair of 3 and 8 at 8,4 and 9,4. And here’s a
little magic for you: In that box we have discovered that the 3 or
the 8 must go in either of the squares at 8,4 or 9,4 although we
don’t know which way around. But if that’s true, then these two
squares must be the only two squares to contain these numbers in
this column as well. So we now know that the 8 in that column cannot
be at 2,4 with the 5. That only leaves the 5 in that square, so it
is solved and as a consequence the 8 in box 2 must be at 2,5.
What we discovered was a matched pair, and these can help to solve
the stickiest of problems. Sometimes you might see a pair such as 3
8 9 and 3 8 in a box, row, or column. If the 9 is somewhere else in
the options of that element (it would have to be, otherwise it’s the
9 for that element) then you can remove it from the 3 8 9 group.
Why? Because we know that those two squares are the only squares for
a 3 or an 8: If a 3 is in one square and an 8 in the other there’s
no room for the 9. In the lingo, that’s called a hidden matched
pair. In Figure 1-15 our matched pair eliminated just one 8, but
sometimes such a matched pair can get rid of large numbers of
extraneous options. 
Figure 1-15. A trio in column one: three
squares sharing three numbers exclusively.
Probably the most difficult construct to
get your head round is a step up from matched pairs where three
numbers share three squares in a box, row, or column. The same
principle applies: The three squares must contain the three numbers
exclusively. For example, if the three numbers are 2, 5, and 9 they
may appear in the options of an element as, say, 2 5, 5 9, 2 5 9 or
2 5, 2 5 9, 2 5 9, or simply 2 5 9, 2 5 9, and 2 5 9.
Looking at the three squares 4,1, 7,1,
and 8,1 in Figure 1-16 the three sets of numbers there all belong to
a group of three numbers that are 5, 8, and 9. They follow the rule
of three numbers and only those numbers in three squares. The result
of applying that rule is that anywhere else in that column the
options 3, 5, or 8 may be eliminated. Goodbye 8 at 1,1 and we have
solved two squares shown in Figure 1-16: 2 at 1,1 and 8 at 1,2. Once
again, for clarity’s sake our example only eliminates a single
option, but usually many more may be culled using this method.
Figure 1-16. Eliminating the options.
The strategies for solving su doku thus
far allow you to solve all but the most difficult and extreme su
doku. The more difficult constructs and
strategies are best learned by practice. The more su doku you solve
the easier it becomes. Twins, triplets, and pairs pop out at you all
over the place. And when they don’t, you’ll be able to find help on
one of the many Web sites and su doku forums that have been set up
just for su doku solvers. Two of the best are
www.sudoku.com and
www.sudoku.org.uk.
Guessing and Ariadne’s Thread
Guessing isn’t only totally unnecessary
for the vast majority of su doku puzzles, but can often land you in
deeper trouble than you were before you started to guess.
For example, you have a number of
squares that contain two options and you can’t see any logical
moves. So you pick a square, choose one of the numbers and carry on
solving using that number. One of three things might happen:
-
You could be very lucky and come to
a final solution.
-
The path you take might come to
another sudden stop.
-
You find yourself in a position with
two same numbers in an element.
In either of the last two scenarios
you’d have to return to your starting point and either pick the
other number or go to another pair and pick one of them. This
trial-and-error method will eventually result in a solution, though
it may take a while.
(The
above is an extract from Su Doku for Dummies)
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