Sudoku strategies richard dickerson

SUDOKU STRATEGIES
AND THE SUDOKERSON
MATRIX

RICHARD E. DICKERSON
PASADENA, CALIFORNIA
APRIL 2008
OO

Sudoku Strategies and the Sudokerson Matrix 2

TABLE OF CONTENTS
Preface 3

1. The Sudokerson Matrix 4

2. Level: Easy 8

3. Level: Medium 13

4. Level: Hard 18

5. Level: Diabolical 23

6. Beyond Diabolical 32

7. Over the Top 43

8. Theseus Lives! 49

9. Sudoku Strategies 50

10. A Field Test of Parallel Analysis 53

Appendix 1: Notes on Sudoku Design 56

Appendix 2: How Many Different Sudoku Puzzles Are There? 61


PREFACE
This book had its origin in the fall of 2006, and was intended originally as a
Christmas gift for two grandchildren who had become fascinated by Sudoku. When it
was begun, I had encountered Sudoku puzzles only in newspapers. I had never heard
of "X-wings", or "swordfish", or "naked pairs" (at least not in this context) or "jellyfish".
Hence the book represents what I found to work best by trial and error. In retrospect it
turns out to be quite different from other books on Sudoku strategy, but I still prefer it
for its simplicity and its power. To date I have never found a published Sudoku puzzle
that could not be solved in a straightforward manner by the four-step methods in this
book. They also are powerful enough to let you take a Sudoku puzzle apart, see how it
is put together, and create your own variations, some of which can be quite interesting.

The analyses in this book are carried out on what I have modestly called the
"Sudokerson Matrix". This is strictly a pencil-and-paper process, with occasional
recourse to a xerox machine to save time making multiple copies. A friend, Bill Chapin,
showed me how to put the Sudokerson Matrix onto a laptop computer using the Excel
program. This makes life easier and eliminates the need for a photocopier, but it must
be emphasized that the computer is not "solving" the puzzle for you in any sense of the
term. It merely is a more efficient kind of pencil and paper and photocopier; you still
do all of the analysis and all of the solving.

The methods in this book are four in number:

(1) In honor of our local Hollywood culture I have called the simplest strategy the 'Snow
Trouble method, or "Cancel while you work!" This will get you through Sudokus
classed variously as Easy, Gentle or Mild.
(2) The search for twofold, threefold and fourfold ambiguities. This will carry you
onward through Medium or Moderate puzzles.
(3) The search for digits that can only occupy only one cell in a given row, column or
box. This very powerful technique gets you through puzzles rated Hard or
Tough.
(4) The method of parallel analysis. This approach is enough to defeat even those
Sudokus that are classed as Diabolical or Fiendish. It also gives you a fascinating
window into the structure and construction of Sudoku puzzles, and may tempt
you into designing your own.

Of course claiming infallibility for a solution strategy is a Clarion Call for
skeptics to design Super-fiendish-diabolical Sudokus that cannot be solved by these
methods. I have seen a few of these, usually involving substantially fewer starting
digits than is customary. So I will not claim that these methods are all-powerful; I will
only say that to date they have never failed to solve any Sudoku encountered in the
published media.

Enjoy!


1. THE SUDOKERSON MATRIX
Sudoku, as you know, involves a 9 x 9 matrix of cells, which can be thought of
either as (a) nine horizontal rows, (b) nine vertical columns, or (c) nine 3 x 3 boxes.
Starting from a given handful of numbers, the object is to find numbers for the blank
cells so that all of the nine rows, all of the nine columns, and all of the nine boxes, each
contain a complete set of digits from 1 through 9. In a classical Sudoku puzzle, starting
digits are positioned in a symmetrical manner around the center of the matrix, although
two symmetry-related digits do not have to be the same. All of the puzzles in this book
at least approximate this symmetry. Some more recent computer-generated examples
abandon symmetry entirely and distribute digits at random across the matrix, but I
frankly find these less interesting.

The trick in solving a Sudoku puzzle is to keep track of digits that cannot be
allowed in each empty cell, and eventually to determine the sole digit that can occupy
that cell. It is a process of elimination. Playing on the usual small published newspaper
Sudoku layout makes it difficult to keep track of what can, and cannot, be allowed. To
make life easier, I have designed the Sudoku matrix shown as Plate 1.1. (Make Xerox
copies of Plate 1.1 for your own use.) For ease of reference, the nine rows are labeled 1
through 9, and the nine columns are labeled A through I. The central cell, for example,
is cell E5, and the cell in the lower right corner is I9.

Double lines separate the nine 3 x 3 boxes, numbered 1–9 as at the foot of Plate
1.1. Hence for example, cell C9 occupies the lower right corner of box 7, and G1 sits at
the upper left corner of box 3. Each of the 81 cells has nine dots, symbolizing the digits:
1 2 3
4 5 6
7 8 9

A crossed-out or otherwise marked dot within a cell indicates that this particular digit is
not allowed because it clashes with the same digit found elsewhere in the row, column
or box containing that cell. I have found that the quickest and simplest plan is to cancel
dots by marking them with a fine-tipped red felt pen. If • indicates one of the original 3
x 3 dots and ! is a cancellation by marking pen or other means, then the following
array tells you that the only possible digits remaining for that cell are 3 or 8:

! ! •
! ! !
! • !

When you figure out what a given cell must contain, you can then overwrite the correct
digit in black.

Bill Chapin showed me how to display the Sudokerson matrix on a laptop
computer using the Excel program. His version of Plate 1.1 is displayed in Plate 1.2,
which shows how you can place numbers of any size and style wherever you want. In


row 1 are five different sizes of the same digit, and row 2 displays the open-outline
format that I find convenient to identify the clashing digits in a failed trial. Row 3
demonstrates how two through six digits can fit into one cell. A matrix with the
starting digits large and all undecided cells blank can be used exactly as the original
matrix in Plate 1.1.

But there's yet another advantage to the Excel program that will become useful
later. Rows 4, 5 and 6 of Plate 1.2 show a portion of a Sudoku in which the starting
digits are in large type, and each cell that is initially unspecified has all nine possible
digits. Instead of marking a 3x3 array of dots, on a laptop computer one can simply
delete digits that clash with other known digits. To illustrate this, in rows 7, 8 and 9
below it, I have cancelled all the digits that clash with the starting set, as though this
was some kind of mini-Sudoku. Every digit but 4 has been canceled in cell E7, so as a
starting point make that digit full-size and then continue the search.

The Excel matrix is particularly useful in the later stages of solving a Sudoku
puzzle, after more than half the digit possibilities have been eliminated. In the initial
stages, running a felt tipped marking pen down a hard copy of Plate 1.1 is faster and
easier than manipulating the matrix on the computer. But there comes a time when the
choices remaining are few but confusing, and it helps to change from Plate 1.1, which
shows you which digits cannot be used, to Plate 1.2, which can keep track of which
digits are still possible.

The next four chapters describe strategies for solving Sudokus that are rated
Easy, Medium, Hard and Diabolical. For the Easy chapter we will use only the hand-
drawn matrix of Plate 1.1. For the Medium and Hard chapters we will place the
starting digit set in an Excel matrix, but then cross out forbidden digits as before. For
later chapters we will go over completely to the computer deletion of forbidden digits
as in the middle and bottom thirds of Plate 1.2.

Bill Chapin's Excel computer display is an enormous asset when solving the most
difficult levels of Sudoku puzzles. Make no mistake; it is not a puzzle-solving program
per se, since you still have to do all the thinking for yourself. But its ability to add and
remove digits quickly, to enlarge or shrink them, or display them in special formats, is
an immense help in rapidly recording what you are thinking. Because it helps
terminate a Sudoku solution in record time, I call it the Chapinator, after our illustrious
California governor. The advantage of the Chapinator over hand-drawn hard copy
solutions is analogous to the advantage of the printing press over hand illuminated
books. You don't have to use Excel to solve Sudokus by the methods used in this
booklet, but it helps.


Plate 1.2: The Excel Matrix

Col: A B C D E F G H I
Row:

1 2 3
5 5

5 5 5
• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 5 5
4 5 6
• • • • • • • • • •
5• • • 5• • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

45 456 45 456 456
7 8 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • 67 78 789
• • • • • • • • • • • • • • • • • • • • • • •

7 1 9 2 5
• • •
1
•
2 3•
• • • • • • •
1 2 3•
• • • • •
1
•
2• 3• 1
•
2 3•
• • • •

4 • • •
4
•
5 6•
• • • • • • •
4 5 6•
• • • • •
4
•
5• 6• 4
•
5 6•
• • • •

• • •
7
•
8 9•
• • • • • • •
7 8 9•
• • • • •
7
•
8• 9• 7
•
8 9•
• • • •

6
1 2 3•
• •
1
•
2 3•
•
1
•
2
•
3• 1 2 3•
• • • • •
1 2 3•
• •
1
•
2• 3• 1
•
2 3•
•
1 2• 3•
•

5 4 5 6•
• •
4
•
5 6•
•
4
•
5
•
6• 4 5 6•
• • • • •
4 5 6•
• •
4
•
5• 6• 4
•
5 6•
•
4 5• 6•
•

7 8 9•
• •
7
•
8 9•
•
7
•
8
•
9• 7 8 9•
• • • • •
7 8 9•
• •
7
•
8• 9• 7
•
8 9•
•
7 8• 9•
•

1 2 3•
4 8 3 1 9
• • •
1
•
2 3•
•
1
•
2
•
3• • • • • • • • • • • •
1
•
2 3•
• • • •

6 • • •
4
•
5 6•
•
4
•
5
•
6• • • •
4 5 6•
• • • • • • • •
4
•
5 6•
• • • •

• • •
7
•
8 9•
•
7
•
8
•
9• • • • 7 8 9•
• • • • • • • •
7
•
8 9•
• • • •

7 1 9 2 5
• • • •
3 •
• • • • • • • • • • • • • •
3• • •
3 •
• • • •

7 • • • •
6 •
• • • • • • • •
4• • • • • •
4 •6 • •
4 •6 • • • •

• • • •
8 •
• • • • • • • • • • • • • •
8• • •
8 •
• • • •

6
•
2 3
• • •
2• 3 • •
2 3
• • •
1 • • • • • •
1
• • •
2• 3 • •
2 •3 • •
2 3
• •

8 •
5
• • •
5 •
• •
5
• • •
4 •5 • • • • •
4• 5 • •
4
• • •
4 •
• •
4• •

•
8•9 • •
8• 9 • •
8• 9 • •
7• • • • • •
7
• • •
7• 8 • •
7 •8 • •
7 •8 •

4 8 3 1 9
• • • •
2 •
• •
2
• • • • • • • • • • • • • • •
2 •
• • • •

9 • • • •
5• 6 • •
5• 6 • • • • •
5• • • • • • • • •
6 •
• • • •

• • • • • • • • • • • • •
7
• • • • • • • • •
7 •
• • • •


2. LEVEL: EASY
The first strategy by itself is often sufficient for Easy puzzles. If you initially
locate three or four cells that can accept only a single digit each, then filling these in can
generate more single-digit cells, and the process may continue until the entire Easy
puzzle is solved. Whenever eight of the nine dots in one cell are crossed out, then the
ninth one must be the right answer. For example:

! • !
! ! !
! ! !

means that only the digit 2 is possible. Fill that digit in, and then do not forget to cross
out or cancel the same digit all through the row, column and box that contain the cell
in question. ("Cancel while you work!") Do it immediately, just after you fill in the
digit. This is perhaps the most important single rule in Sudoku, since these newly
cancelled digits are what you depend on to give you fresh information that lets other
cells be identified. Canceling along rows and columns becomes a reflex, but it is all too
easy to forget to cross out the newly added number elsewhere within its box as well.

This chapter deliberately avoids using the Chapinator, just to show that it can be
done. Plate 2.1 is an example of a Sudoku puzzle that is rated Easy. Note the
symmetrical arrangement of digits. Cells H1 and B9 are arranged symmetrically
around the center of the matrix, and each contains a digit, although one is 2 while the
other is 6. Cells G4 and C6 also are symmetric, although one has a 1 and the other a 5.
Every digit in the puzzle is matched by another digit (not necessarily the same) in a cell
symmetric to it. This is not an iron-clad rule in Sudoku, but is a style choice that gives
the puzzle a feeling of elegance. (In x-ray crystallography, which is my field, this is
termed either a "center of symmetry" or a "twofold symmetry axis". The two are the
same for figures in a plane such as diagrams on paper.)

The first step in solving any Sudoku puzzle is to consider each of the given
numbers in turn. For a number in a given cell, cross out this same number everywhere
else that it occurs in the row, column, and box that contains that cell. In Plate 2.2 I have
done these initial cross-outs for the 5 in the third row, and the 3 in the sixth row as
examples. Rapid dots with a colored felt pen are the best way of crossing out digits, but
I use small circles here to make overwriting easier to read in the absence of color. It is
convenient to underline a number after you have finished crossing out its forbidden
locations, just to help you keep track.

Plate 2.3 shows the complete number cross-outs or eliminations for the entire
puzzle. As we shall see later with more complex puzzles, it is convenient to refer to the
starting matrix, Plate 2.1, as "sheet A", and the matrix after all initial crossouts have
been made, Plate 2.3, as "sheet B". "Sheet C", to be defined later, is the starting point for
higher-level analysis.


Notice in sheet B that cells A4, B7, F4, G3, H3, I6 and I9 each have only a single
possible digit: 9, 4, 8, 3, 7, 6 and 7, respectively. Any one of these could be the start of
this puzzle. Consider B7, which can only hold a 4. Crossing out this 4 everywhere else
in the row, column and box that contains cell B7 means that B2 must be 9, B8 is 2, and
C7 is 7. With these additions, Box 7 at lower left has obtained all its numbers except 9,
and the only place left for the 9 is in cell C9. Hence you have solved box 7. The moves
just described can be listed economically as:

1) B7=4
2) B2=9
3) B8=2
4) C7=7
5) C9=9
6) A4=9
7) C2=4
8) C5=2 .....and column C is solved.

This approach works only if you are absolutely rigid about deleting a number from
other cells in its row, column and box just as soon as you have used it somewhere. This
cannot be stated too strongly. If you fail to do this, the following listing won't make
sense and you probably won't be able to solve the puzzle. To continue:

9) D7=6
10) F8=7
11) F4=8
12) F9=2
13) F6=1
14) F3=6
15) F1=9 .....and column F is solved.

16) D9=8
17) D6=2
18) D4=5
19) D2=7
20) D1=1 .....and column D is solved.

21) G3=3
22) H3=7
23) A3=1 .....and row 3 is solved.

24) A1=7
25) A5=6 .....and column A is solved.

26) B6=8
27) B5=1
28) B1=3 .....and column B is solved.


29) E9=4
30) E8=3
31) E5=7 .....and column E has only a [5 8] ambiguity at E1 and E2.
32) G8=4
33) G1=8 .....which breaks the just mentioned [5 8] ambiguity.
34) E1=5
35) E2=8 .....and both row 1 and column E are solved.

36) G5=9 .....and column G is solved.

37) H9=1
38) H8=6 .....and row 8 is solved.
40) I9=7 .....and row 9 is solved.
43) I5=5 .....and column I is solved.
45) H2=5 .....and row 2 and column H are both solved.

In fact, the entire puzzle has been solved, as seen in Plate 2.4.

Sudoku puzzles don't get much simpler than this. All that you had to do here
was to look for single-digit cells, be religious about immediately canceling the newly
added digit in row, column and box, and then look for new single-digit cells. That
single-idea approach won't get you very far in puzzles that are classified as Medium,
and still more subtlety is needed for those labeled Hard and Diabolical. We shall
consider a Medium puzzle next.


3. LEVEL: MEDIUM
A very powerful tool is the recognition that, if two cells in a given row are each
limited to the same two numbers, then that uses up these numbers for the row, and
both numbers can be crossed out wherever else they occur in that row. If a and b
represent digits, and square brackets [ ] indicate different cells, then the twofold
ambiguity rule can be written as:

[a b] [a b] —> Neither a nor b can appear in any of the other seven cells
of that row.

What has been said, and will be said, about rows, applies equally well but
independently to columns and to boxes. For example, if one 3 x 3 box has two cells that
each are limited to either a 5 or a 7, then 5 and 7 can be deleted from all the other seven
cells of that box. The essential criterion for using these twofold ambiguities to cross out
digits is that the two cells should be absolutely identical. Allow the possibility of a third
digit in one of them, and the method fails. But the method of twofold ambiguities
works independently for rows, for columns, and for boxes, making it very powerful.

Threefold ambiguity cancellations also are encountered frequently. If three
digits, a, b and c, appear in some combinations in three different cells [ ] of a row (or
column, or box), and if no other digit is allowed in these three cells, then these three
cells must contain a, b and c in some as yet-undetermined order. Hence a, b and c can
be crossed off as forbidden to the other six cells of the row (or column, or box).
Examples of threefold ambiguities are:

[a b c] [a b c] [a b c] This one is less common but does turn up.
[a b c] [a b c] [a b]
[a b c] [a b] [a c]
[a b] [b c] [c a] This one is reasonably common.

But a fourth digit in one cell wipes out the method entirely, as:

[a b d] [b c] [c a] No conclusions can be drawn.

For if the first cell holds a d, then either a or b would be forced to show up in one of the
remaining six cells in the row, and you couldn't generalize and cross out a, b and c from
those cells.

Fourfold ambiguity cancellations are less common, but occur more often than
you might think. If four different cells contain only two or more of a set of four digits,
then these four digits can be deleted from the other five cells of that row, or column or
box. It turns out after the fact that this strategy is a familiar one to Sudoku solvers. For
example, Andrew C. Stuart in his "The Logic of Sudoku" (Michael Mepham, 2007) refers
to twofold, threefold and fourfold ambiguities as naked pairs, naked triples and naked
quads.


Let's try out the twofold ambiguity strategy with a Sudoku puzzle rated
Medium, on sheet A (Plate 3.1). We will switch to the Excel form of the matrix because
it is cleaner, but will continue with manual crossing-out for the moment.
Row/column/box cancellation of all of these original numbers leads to sheet B (Plate
3.2), and this is where the puzzle really begins. Sheet B reaches an impasse at once
using just the methods of the Easy puzzle. There are no initial single-digit cells at all.
But use of twofold and higher ambiguities will lead to a solution. The path of analysis
is listed below. (As before, whenever you assign a number to a cell, then immediately
cancel that number everywhere it occurs in the row, column and box that contains that
cell.)

1) No single-digit cells are present after initial crossouts are completed.
2) D6/D7=[3 5]. That is, both D6 and D7 constitute an ambiguous pair.
Either cell can hold a 3 or a 5, but nothing else.
Cancel all 3's and 5's elsewhere in column D only.
3) D2=6
4) A2/A5=[3 4]. Cancel other 3's and 4's from column A.
5) A9=8, A7=9, A1=6, D8=1, D5=2
.....and column D is solved except for the [3 5] twofold ambiguity at
D6/D7. But this ambiguity doesn't give us any new information
because we already know the other seven cells in column D.

6) F3/H3=[3 5]. Cancel all other 3's and 5's in row 3.
7) E3=2, I3=6, B3=9
.....and row 3 is solved except for the [3 5] ambiguity at F3/H3.

8) F4/F5=[3 4]. Cancel other 3's and 4's in column F and in box 5.
9) F3=5, so H3=3, breaking one ambiguity and solving row 3.
10) F8=8, E2=3, so F2=7
.....and column F is solved except for the [3 4] ambiguity at F4/F5.

11) H4=4, so F4=3, F5=4, A5=3 and A2=4, breaking two ambiguities.
.....and columns F and A are solved.

12) E5=1, I5=7, and row 5 is solved.

13) I1/I8=[4 5] ambiguity. Cancel other 4's and 5's in column I.
14) So I9=3, G9=4, G1=7
15) G8=9, G6=3, G4=6, and column G is solved.

16) I8=5, so I1=4, breaking the I1/I8 ambiguity and solving column I.

17) H1=5, C1=3, B1=2, completing row 1.
18) C2=5, B4=1, C4=9, completing rows 2 and 4.
19) C6=4, so B6=7


20) E8=4, so E7=5, completing row 8 and column E.
21) D7=3, D6=5, completing column D
22) B7=4, H7=7, completing row 7.
23) H6=8, B9=5, C9=1, H9=2
.....and the entire puzzle is solved, as shown in Plate 3.3.

Note that this puzzle would have been totally unsolvable were it not for the twofold
ambiguity strategy, whereas the previous Easy puzzle didn't need it at all.

But these methods are still insufficient enough to solve a really Hard puzzle.
Another strategy remains to be invoked, and it will be used next with a puzzle rated as
Hard.


Plate 3.1: Initial digit set, A

Row:

9 8 1
• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 2 1 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 7 4 8
• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 8 7 2
• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 8 5 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 6 9 1
• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 2 1 8
• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 3 2 6
• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 9 6
• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •


Plate 3.3: Final solution, C

Row:

6 2 3 9 8 1 7 5 4
• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 8 5 6 3 7 2 1 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 9 7 4 2 5 8 3 6
• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 1 9 8 7 3 6 4 2
• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 6 8 2 1 4 5 9 7
• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 7 4 5 6 9 3 8 1
• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 4 6 3 5 2 1 7 8
• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 3 2 1 4 8 9 6 5
• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 5 1 7 9 6 4 2 3
• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •


4. LEVEL: HARD
The two previous puzzles could be solved without using a powerful technique
which I call "digit searching". Up to this point the main question has been: "Which
digits can be allowed in any given cell?" Now we will reverse this and ask: "Which
cells can still accommodate a given digit?" The former strategy will work with most
puzzles rated Easy or Medium. But for Hard puzzles, the new strategy must be added.

To demonstrate this, the puzzle in sheet A (Plate 4.1) is rated Hard. Simple initial
mapping of disallowed digits leads to sheet B (Plate 4.2). The puzzle began with 26
specified digits, and use of single and double ambiguities as in the earlier examples
leads to identification of seventeen more digits, by the process:

1) D9=9, B9=8, A9=7
2) A7/A8=[5 9] ambiguity.
Delete other 5's and 9's from column A and box 7.
3) A3=8
4) A2/A6=[1 2] ambiguity. Delete 2 from A4.
5) A4=6
6) F1/F6=[5 9] ambiguity. Delete other 5's and 9's from column F.
7) F5=7, F3=6, F8=8
8) A6/H6=[1 2] ambiguity. Delete other 1's and 2's from row 6.
9) G6/I6=[4 9] ambiguity.
Delete other 4's and 9's from row 6 and box 6..
10) F6=5, so F1=9, completing column F.
11) D5=2, so D4=4, D7=5 and D2=3, completing column D.
12) E5=9, A7=9, so A8=5

This stage of the analysis is shown in Plate 4.3. The analysis has stalled with thirty-
eight cells remaining unsolved. Where do we go from here? New methods are needed.

It cannot be emphasized too strongly that whenever you add a number to the
column, you must be religious about immediately crossing that number out in the
relevant row, column and box. These crossouts are your sources of new information,
and in their absence the solution will go astray.

As an example of the importance of immediate cancellations, three digits in
Plate 4.3 that should have been cancelled have not been. Without them, some steps
in the continued analysis below make no sense. (Try them and see.) Find the
missing cancellations by starting with Plate 4.2 and carefully carrying out steps 1 - 12
listed above. As a clue, look carefully at box 6. Or if all else fails, check the end of
this chapter. And then see what these added cancellations do for the subsequent
analysis, steps 1 - 19.


The new strategy of this chapter is systematically to examine rows, columns and
boxes for missing digits that have only one possible location. (Hence: "Which cells will
a given digit fit into?") In column A of corrected Plate 4.3 there are two places for a 1
and two (the same two) for a 2, so one learns nothing. Column B lacks 2, 3, 4, 5 and 9,
but there are at least two possible locations for each of them, so again nothing is
learned. In fact, columns A through F all offer no new information. In column G, digits
1, 3, 4, 6, 8 and 9 are yet to be added, but only 4 has a unique position, at cell G6. In the
abbreviated listing of the previous two puzzles, this search of Plate 4.3 begins as below.
The expression:

column G: (1 3 4 6 8 9) G6=4

will be taken to mean that the digits 1, 3, 4, 6, 8 and 9 have not yet been assigned in
column G, but that all of them have two or more possible locations except for 4, which
is limited to cell G6. Because this "Finding cells for digits" method is so powerful
compared to methods we have used so far, all entries below using this new strategy
have been italicised.

BEGINNING WITH COLUMNS A - I:

1) columns A - F no new information.
(Cols. D and F have already been solved.)
2) column G: (1 3 4 6 8 9) G6=4
(This is a key development, that exists only because no other
cell in column G can accept a 4.)
3) column G: (1 3 6 8 9) G8=9
(We just looked at column G, but putting a 4 in cell G6
now leaves only G8 available for a 9.)
4) column H: (1 2 3 5 6 8) H7=8
5) column I: (1 2 3 4 6 7 9) I7=7

TURNING NOW TO ROWS 1 - 9:

6) row 1: (1 2 4 5) E1=2
7) row 2: (1 2 5 6 7 8) G2=8
8) row 6: (1 2 9) I6=9
9) The other rows all have multiple sites for missing digits and contribute
nothing.

EXAMINING 3 X 3 BOXES 1 - 9:

10) box 3: (1 3 4 5 6) I2=6
11) box 5 is solved, and boxes 2 and 7 have twofold ambiguities that are
of no help because all the other 7 cells in that box are known.
12) boxes 1, 4, 6 and 8 have no unique sites for any of their missing numbers.


COLUMNS REEXAMINED

13) Re-inspection of columns A - H yields no new data.
14) column I: (1 2 3 4) I1=1

Note that E7/E8/E9 form a triple ambiguity with digits: [1 4][4 6][1 6], telling us
that none of the remaining six cells in column E and none of the other six cells in box 8
can contain a 1, 4 or 6. But we already knew that.

15) H1=5, so C1=4
16) H3=3, so I3=4

ROWS REEXAMINED

17) row 1 is complete and rows 2 - 7 and 9 yield no new data.
18) row 8: (2 3 4 6) E8=4, so E7=1 and E9=6

THE FINAL CASCADE

19) At this point, with 24 cell still unassigned, a veritable explosion of new
decisions follows, as each new addition produces one or more new candidates:
G9=1, G7=3, I8=2, I4=3, H8=6, H5=1, H6=2, G5=6, C8=3,
B7=4, C4=9, B4=2, A6=1, A2=2, B2=5, B3=9, B5=3, C5=5,
C3=7, C2=1, E3=5, and E2=7.
Without further ado, the entire problem is solved as shown in Plate 4.4.

This is an enormously powerful strategy. The best plan is to search
systematically down rows 1 – 9, then columns A – I, then boxes 1 – 9, inserting newly
located digits and crossing these digits off elsewhere. Whenever this produces an
ambiguous 2-fold, 3-fold or 4-fold set of cells, or simply limits a cell to one digit, make
this change and continue. Repeat this cyclic process until a cycle goes by without
addition of a single new digit. If all goes well you will have solved the puzzle by this
point.

(Corrections needed in Plate 4.3: Cancel the 1 in cell G6, and 1 and 2 in cell I6.)


Plate 4.1: Initial digit set, A

Row:

3 6 8 7
• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 2
• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 1 5 7
• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 8
• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 8 6 3
• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 2
• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 7
• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 3 4 5
• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •


Plate 4.4: Final solution, C

Row:

3 6 4 8 2 9 7 5 1
• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 5 1 3 7 4 8 9 6
• • • • • • • • • • • • • • • • • • • • • • • • • • •

2 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 9 7 1 5 6 2 3 4
• • • • • • • • • • • • • • • • • • • • • • • • • • •

3 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 2 9 4 8 1 5 7 3
• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

4 3 5 2 9 7 6 1 8
• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

1 7 8 6 3 5 4 2 9
• • • • • • • • • • • • • • • • • • • • • • • • • • •

6 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 4 6 5 1 2 3 8 7
• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

5 1 3 7 4 8 9 6 2
• • • • • • • • • • • • • • • • • • • • • • • • • • •

8 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •

7 8 2 9 6 3 1 4 5
• • • • • • • • • • • • • • • • • • • • • • • • • • •

9 • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • • •


5. LEVEL: DIABOLICAL
The methods described so far will be referred to collectively as the "three basic
strategies". These will get you a long way, but they still usually fail with puzzles that
are rated Extremely Difficult or Diabolical. The basic strategies alone can halt at a stalled
solution, dead in the water, with no suggestion as to what to do or where to go next. A
new and more powerful procedure is needed. This is the method of parallel analysis,
and it will turn out to be the strongest weapon in the Sudoku-solver's arsenal. I have
encountered no published Sudoku puzzle yet that could not be solved by the parallel
analysis method. All that is needed is pen or pencil and a photocopier machine. But
here is where Bill Chapin's matrix-drawing Excel file comes into its own; it is easy and
rapid to use, makes the photocopier superfluous, and takes all the pain out of the most
complicated of parallel solutions.

To illustrate the parallel analysis method, the initial Plate 5.1 (which I call sheet
A) contains an Extremely Difficult or Diabolical Sudoku. We now have switched over to
my own extension of the Chapinator file, in which each of the 81 cells begins with a 3x3
grid of digits 1 – 9. The starting set of digits is introduced, enlarging them for
convenience as in sheet A. This approach replaces the 3x3 grid of dots representing
digits that have been ruled out, by a 3x3 array of digits that still remain possible for
each particular cell. Instead of marking the dot that represents a forbidden digit, you
now simply delete that digit from the cell. If you are not using the Chapinator it still
pays to prepare a hand-drawn version of a page like Plate 5.3, so one can make as many
photocopies as necessary and cross out numbers by hand. But the Excel version is much
easier to use.

In solving Plate 5.1, as usual, first delete all digits that are disallowed by the
starting digit set, to obtain Plate 5.2 (which will be termed sheet B). Then use the three
basic strategies seen earlier to go as far as possible toward a solution. If the analysis
goes to completion, you have won. But with Diabolical puzzles there usually come a
time when none of the basic strategies have anything more to offer. What we will call
sheet C (Plate 5.3) displays the solution carried as far as one can go using the three basic
strategies. It is the end of the line for the methods presented so far. The analysis is not
wrong; it simply has stalled with nowhere left to go. The challenge of this chapter is
how to use parallel analysis methods to break the stall observed in sheet C.

Note that Plate 5.3 has nineteen different cells that are limited to two possible
digits each. The starting point in parallel analysis is to select one such cell and try each
of its two possible digits in turn in a continued analysis using the basic methods of the
previous chapters. Not all of these binary cells in Plate 5.3 are genuinely independent.
Cells B6 and B8 are an ambiguous pair that can hold only a 4 or a 5, so when you have
specified the contents of one cell you also know the contents of the other. Other binary
cells on Plate 5.3 are linked as triplets or even quadruplets:


Single cells (7): A9 G5 H6 H8 H9 I8 I9
Paired cells (1): B6/B8
Triplets (2): A2/C2/C6 D3/D7/F7
Quadruplets (1): E2/E6/F5/G2

Hence there are only eleven truly independent two-digit cells, and eleven different
starting points for parallel analysis.

To illustrate the method in its simplest form, let us consider cell H9, and try its
two options: 1 or 7. Setting H9=1 and proceeding with the basic strategies leads to the
crash shown in Plate 5.4, with two conflicting 2's in row 5 and two 7's in row 6. But
things go better with H9=7: the puzzle is solved correctly in Plate 5.5.

(A word of warning about crashed solutions. There is quite a bit of play or
freedom in arriving at an incorrect solution. If you repeat this and the following
analyses on your own, you may find that your crashed solutions are different in many
details from my Plates 5.4, 5.6 or 5.8. But they will all exhibit the same kind of faults:
duplicate digits in one or more rows, columns or boxes. In contrast the correct answer,
in Plate 5.5 or 5.9, is unambiguous. There's a moral here. In Sudoku, as in real life,
there are many wrong answers but only one right answer.)

A trial with a chosen digit can have any one of three outcomes. It can:
(@) lead to the correct answer,
(x) crash with duplicate digits in one row, column or box, or simply
(s) stall once more without reaching any conclusions.
If we call the two digits in a chosen test cell Y and Z, then the possible outcomes for its
tests can be:

Y Z Conclusions
@ x The best outcome of all; digit Y is right and Z is wrong. Puzzle solved.
@ s Digit Y is right but Z is ambivalent. There may be another solution
lurking somewhere (but almost never in a published Sudoku).
s x You at least know that Y is right, even though the puzzle isn't yet solved.
s s Nothing new has been learned about either digit choice.
x x You have made a mistake somewhere. Either Y or Z must be right.
@ @ Two independent correct solutions: one with Y and one with Z.
You won't find this situation in a published Sudoku, as designers
detest such an outcome. But see the next chapter.

If either choice of digits solves the matrix the battle is over. If both choices stall you
have obtained no new information. But if one choice stalls while the other one crashes,
you still have learned something. The digit that led to a stall is correct even though it is
not sufficient by itself to solve the puzzle. So add this new digit to the matrix, choose
another binary cell and try again.


It might seem intuitive that the best strategy is to choose a cell that is linked to
other cells, such as E2/E6/F5/G2 in this example, because choosing the digit in one of
these cells also decides the digits in all the others of the set. This generally is true but
not always. If you use E2=4 and its quadruplet set here, the correct answer comes up
immediately. E2=7 leads to a crash. (Try this yourself.) But the D3/D7/F7 triplet is not
so cooperative. A test with D3=4 leads to a crash with duplicate 1's and 4's (Plate 5.6).
The other choice, D3=6, does not solve the puzzle, it only leads to another stall as seen
in Plate 5.7. However, you still have gained new information: you know that D3=6 is
correct even though it is not strong enough to bring about a solution. So add D3=6 to
the matrix and choose another cell for testing.

In this example I added cell B6, which can hold either 4 or 5. B6=5 crashes once
again, in my trial on Plate 5.8 yielding duplicate 2's and 7's in rows 6 and 5. (As
mentioned earlier, if you try this you may wind up with different duplicate digits. But
there will still be a crash.) In contrast B6=4 solves the puzzle (Plate 5.9). And this
solution, as it must be, is identical to that seen earlier in Plate 5.5.

These two solving processes can be mapped by the tree diagrams shown in Plate
5.10. The term pal, for parallel analysis levels, will be used to record how many levels
were needed to solve the puzzle. The trial with cell H9 required only one level of
analysis to produce one solution and one crash, or pal=1. In contrast, the trial with D3
required two levels of analysis (pal=2) to yield the right answer. These two trees may
seem trivial, but other especially tough Sudokus can require three or occasionally even
four or more levels of analysis. In such cases a tree diagram is absolutely essential for
keeping track of where you are. But in most cases the process of parallel analysis is fast,
easy, and straightforward, involving only one or two levels. Diabolical puzzles simply
are no longer diabolical if parallel analysis is used.

Parallel analysis so far has been described as studying alternative choices in cells
that have two possible digits. Of course, there is nothing sacred about two digits. In
Plate 5.3 you could just as well have decided to try out the 4, 6 and 7 that are
possibilities in cell F3. Sometime a threefold choice is good strategy. But it carries one
fundamental danger. With a twofold choice cell, [3 5], if using 3 leads to a crash and 5
to a stall, you have still learned something: the correct occupant of that cell is 5. But
with a threefold cell, [3 5 6], if 3 crashes but both 5 and 6 merely stall, you haven't
learned much. Either 5 or 6 must be correct, but you don't know which one. I have
found that testing twofold cells is generally the better strategy, especially those that are
combined with other cells into pairs, triplets or quadruplets.

In sum, parallel solution analysis is the most powerful Sudoku technique that I
know of. It also can be hypnotic. Ferreting out all the different crashes and wrong
answers, comparing them with one another and with the right answer, can be more
interesting than merely going for the correct solution. If you continue examining
solutions of Plate 5.3 on your own, you will find that it has a few more surprises to
offer; surprises related to the example in the next chapter.

Sudoku strategies richard dickerson

Sudoku strategies richard dickerson

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