GENERAL ¨ ARTICLE

Sudoku Squares as Experimental Designs
Jyotirmoy Sarkar and Bikas K Sinha

Sudoku is a popular combinatorial puzzle. We
give a brief overview of some mathematical features of a Sudoku square. Then we focus on interpreting Sudoku squares as experimental designs in order to meet a practical need.
(left) Jyotirmoy Sarkar is
a Professor and Statistics
Consultant at Indiana
University-Purdue
University, Indianapolis.
His research includes
applied probability,
mathematical statistics and
reliability theory.
(right) Bikas K Sinha
retired as Professor of
Statistics from Indian
Statistical Institute,
Kolkata. He has done
theoretical and applied
research in many topics in
statistics.

Keywords
Adjacency matrix, internal block,
Latin square, orthogonal Latin
squares, orthogonal Sudoku
squares, Sudoku graph, Sudoku
puzzle.

788

1. Sudoku Squares and Sudoku Puzzles
The (standard) Sudoku square of order 9 entails a 9 × 9
array of numbers 1 through 9 such that every row, every
column and every 3 × 3 internal block contains each
number exactly once. Sudoku squares of any other order
n = pq are also similarly defined as an n × n array of
numbers 1 through n such that every row, every column
and every p × q internal block contains each number
exactly once. The internal blocks are generally shown
as ‘bordered’ squares/rectangles of specified order(s).
A well-designed Sudoku puzzle consists of a partially
filled array (the filled values are called clues) which can
be extended to a complete Sudoku square in a unique
way using pure logic and requiring no guesswork. To
solve the puzzle, the player must find this complete Sudoku square solution (as quickly as possible).
We sketch a broad-stroke history of the Sudoku puzzle.
In the late nineteenth century, a version of the puzzle
appeared in French newspapers such as L’Echo de Paris,
but it had disappeared by the time of World War I. The
modern Sudoku puzzle first appeared under the name
‘Number Place’ in the New York puzzle magazine, Dell
Math Puzzles and Logic Problems. In 1984, a Japanese
puzzle company Nikoli introduced the puzzle under the
name ‘Suji wa dokushin ni kagiru’ (meaning ‘the digits must be single’), which was shortened to Sudoku.

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

In 1997, Wayne Gould, a retired Chief District Judge in
Hong Kong discovered a puzzle while visiting Japan. He
began writing a computer program to produce more puzzles. He sent a puzzle to The Times in London, where
it was first published in November 2004, and continues
to be published ever since. By May 2005, the Sudoku
craze had spread around the world and puzzles had begun to appear in newspapers in Australia, Canada, Israel, Eastern Europe, India, and eventually the United
States. Today, the Sudoku puzzle has become a regular
feature in many newspapers and magazines all over the
world, inviting readers from all walks of life to solve this
entertaining and addictive combinatorial puzzle. Many
websites and computer applications offer the puzzle electronically on hand-held devices, making it accessible at
all times in all places.
Sudoku puzzles are classified in increasing order of difficulty as easy, medium, hard, very hard and evil. Alternatively, the Sudoku puzzlers may be categorized as
novice, player, expert and master. The likelihood of
solving a puzzle and the time to solve it depends on the
expertise and diligence of the player. Some websites give
hints based on the next logical step a computer program
would take in solving the puzzle.
Figure 1 shows an example of a Sudoku puzzle that appeared in [1].

Figure 1. The daily Sudoku
on the day this article was
written.

RESONANCE ¨ September 2015

789

GENERAL ¨ ARTICLE

Figure 2. The previous day’s
puzzle and its solution.

Typically, Sudoku solutions are available the next day
or in the next issue of the magazine. For instance, we
found the previous day’s puzzle and its solution in [2]
(Figure 2).
Although not required, in most Sudoku puzzles, the cells
in which values are already given exhibit a half-turn
(180o rotation) symmetry! See Figures 1 and 2.
It should be pointed out that the Sudoku solution only
shows that the puzzle is a legitimate one. It helps neither
the solver nor the player who failed to solve it, unless
the solution includes the order in which the numbers
are filled in. Such information seems to be missing in all
published solutions the authors have come across.
1

For more information on Magic
Squares, see articles at Resonance, Vol.11, No.9, pp.76–89,
2006, Vol.12, No.10, p.79, 2007
and Vol.15, No.8, pp.733–736,
2010.

790

Drawing a parallel to Magic Squares1 (where numbers
1 through n are arranged in an n × n array in such a
way that the sum in each row, in each column and along
each diagonal is a constant n(n2 +1)/2), one may inquire
whether the numbers along the diagonals of a Sudoku
square are all distinct. In general, most Sudoku squares
do not have distinct elements along its main diagonal or
alternative diagonal. See Figure 2, where the main diagonal has (7, 5, 8, 6, 7, 8, 9, 4, 7) and the alternative diagonal has (5, 3, 7, 3, 7, 1, 6, 2, 8). Also, in the Sudoku
puzzle in Figure 1, the alternative diagonal already has
two 8’s.

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

Figure 3. A diagonal Sudoku
square of order 9.

Needless to say that it requires a special care to construct Sudoku squares in which the main or the alternative diagonal contains distinct elements. In the rare
cases when both the main diagonal and the alternative
diagonal contain distinct elements, we call the Sudoku
square a diagonal Sudoku square. Figure 3 exhibits a
diagonal Sudoku square.
A p2 × p2 Sudoku board can be depicted as the Sudoku
graph of rank p by letting each of the p4 cells (numbered
1 through p4 row-by-row) correspond to a vertex, and
by connecting two vertices by an edge if and only if the
cells that they correspond to are in the same row, or
column, or internal block. The adjacency matrix of the
Sudoku graph of rank p is a p4 × p4 matrix A = ((aij )),
with ai,j = 1 if vertices i and j are connected, and 0
otherwise. It can be shown that for all p ≥ 2, the eigenvalues of the adjacency matrix of the Sudoku graph of
rank p are integers [3].
Furthermore, one can assign a distinct color to each
number 1 through p2 , and color all vertices of the Sudoku graph of order p that correspond to cells containing
a given number with the color assigned to that number.
Then, a Sudoku square becomes equivalent to a coloring of the corresponding Sudoku graph such that no two
adjacent vertices have the same color, and solving a Sudoku puzzle becomes equivalent to completely coloring
a partially-colored Sudoku graph.

RESONANCE ¨ September 2015

791

GENERAL ¨ ARTICLE

A typical Sudoku puzzler need not worry about the following mathematical questions which arise naturally:
• How many Sudoku squares are there? (66709037520210
72936960 ≈ 6.67 × 1021 of order 9 [4]. Also, if the symmetries due to the permutation of numbers, bands (top,
middle and bottom sets of three rows), rows within a
band, stacks (left, middle and right sets of three columns),
columns within a stack, and symmetries due to reflections and rotations of the entire square are not differentiated, then there are essentially 5,472,730,538 distinct
Sudoku squares of order 9 [5]).
• How many solutions does a given Sudoku puzzle have?
(As we mentioned before, a well-designed Sudoku puzzle has a unique solution. But, in general, it is very
hard to determine (without the aid of a computer) if an
arbitrary Sudoku puzzle has a solution; and if it does,
whether the solution is unique or there are several solutions. However, the problem was completely solved – a
method was developed to calculate the number of ways
in which one can color the vertices of a partially colored
graph in such a way that no two adjacent vertices have
the same color [6]).
• What is the fewest number of entries that must be
given so that the completed Sudoku square is unique?
(The answer is 17 [7]; or, ≤ 18, if we further require that
the cells with given clues exhibit a half-turn symmetry.
Also, see [8] for almost 50,000 distinct uniquely solvable
Sudoku puzzles of order 9 with 17 clues).
• How is the level of difficulty of a Sudoku puzzle determined? (This can be quite subjective. Oftentimes,
the difficulty rating depends on the relevance and the
location of the clues rather than the number of clues.
A computer program can determine the difficulty for a
human player based on the complexity of the required
solving techniques [9]).

792

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

However, the puzzler can benefit from answers to the
question: What are some strategies for solving a Sudoku
puzzle of a given difficulty level? We will not address
this important question in this article, except to advise
the interested puzzlers to search the Internet using key
phrases such as ‘How to Solve a Sudoku?’ and ‘Math
Behind Sudoku’. Hopefully, the puzzlers will arrive at
helpful sites such as [10] and [11].
In this article, we will focus on using Sudoku squares as
statistical experimental designs in order to meet some
practical constraints. But no knowledge of statistics is
needed to understand this article.
2. Sudoku Square as a Specialized Latin Square
Every Sudoku square is a special kind of a Latin square2
(where numbers 1 through n are arranged in an n × n
array such that every row and every column has each
number exactly once). In fact, Sudoku squares form
a tiny proportion of Latin squares of the same order.
For example, there are about 5.525 × 1027 Latin squares
of order 9, which is about 828,000-fold bigger than the
number of Sudoku squares of order 9 [12].

2
Also see Resonance, Vol.17,
No.9, p.895–902, 2012.

Latin squares are used as statistical experimental designs that study the effect of three explanatory variables,
each at n levels, on a response variable, using only n2
experimental units. In a Latin square design (LSD), the
experimental units are laid out in an n × n square array. The Row (R) represents one explanatory variable,
the Column (C) represents a second explanatory variable, and the Treatment (T ), shown as a number in the
(i, j)-th cell of the square, represents a third explanatory
variable.
Note that, in an LSD, the roles of R, C and T can be interchanged. Hence, we can say that every row and every
treatment has each column exactly once; or that every
treatment and every column has each row exactly once.

RESONANCE ¨ September 2015

793

GENERAL ¨ ARTICLE

Figure 4. Various representations of the same Latin
square design of order 5:
(i) T within (R uC),
(ii) C with-in (R uT), and
(iii) R within (T u C).

See Figure 4 for three representations of the same LSD,
even though the Latin squares appear to be different.
Whenever a Latin square is usable as an experimental
design, one can instead choose a Sudoku square of the
same order as the experimental design. This will allow incorporating a fourth explanatory variable (internal block) without increasing the number of experimental
units, and thereby provide an added practical benefit.
For example, a standard Sudoku square of order n = 9
also forces every 3 × 3 internal block to contain each
number exactly once (just as every row and every column does). Which additional variable can we identify
with internal block?
Following [13] and [14], we discuss the use of a Sudoku as
an experimental design in the context of a manufacturing example in which the Row (R) represents the suppliers (S) of raw material, the Column (C) represents the
days (D) of operation, the Treatment (T ) is identified
with Machines (M) used in the operation, represented
by the numbers 1 through 9, and the Internal Block (B)
is identified with Operators (O), also represented by the

Figure 5. A Sudoku of order 9
with one operator (O) assigned to each internal block.

794

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

numbers 1 through 9. See Figure 5. The identification of an internal block as a fourth explanatory variable
was a simple yet profound achievement that gave a new
life to the otherwise artificial requirements of a Sudoku
square. A Sudoku-based statistical experimental design
was born, which we now call a Sudoku design. With
reference to the above example, the Sudoku of order 9
allows some flexibility on the availability of the 9 operators, who are available as teams of three for 3 days
only: Operators O1, O2 and O3 will work on Days 1-23, Operators O4, O5 and O6 will work on Days 4-5-6,
and Operators O7, O8 and O9 will work on Days 7-8-9.
3. Balanced Incidence Matrix and Orthogonality
of Variables in a Sudoku Design
Consider the four explanatory variables in a Sudoku design: Row (R), Column (C), Internal Block (B) and
Treatment (T ). The incidence matrix for any two of
these four variables is a two-way display of the number of units that receive each level combination of these
two variables. Since there are exactly n2 units in total, the two variables each with n options are said to be
perfectly balanced if their incidence matrix is an n × n
matrix with all entries 1; that is, the incidence matrix is
Jn = 1n 1
n , where 1
n = (1, 1, . . . , 1) is an n-component
vector of unity; that is, every level combination of the
two variables occurs in exactly one experimental unit.
The combinatorial requirement of perfect balance between two variables translates into these two variables
becoming orthogonal to each other. Orthogonality of
two variables means that the effect of one variable can
be measured without it being contaminated by the effect
of the other variable. Without orthogonality, measuring
effects of variables may be impossible, or difficult and
less precise. So, orthogonality is a very desirable statistical property. For further details, we refer the readers
to any standard statistics textbook such as [15]. For the

RESONANCE ¨ September 2015

795

GENERAL ¨ ARTICLE

Figure 6. Operator assignment in the Sudoku of Figure
5.

purpose of this article, it suffices to interpret orthogonality between two variables as the combinatorial property
of perfect balance between them.
We have already mentioned (in Section 2, paragraph 1)
that a Latin square design exhibits row–column, row–
treatment and treatment–column orthogonality. What
about the orthogonality between any two variables in
a Sudoku design, where we also have internal blocking
as a fourth explanatory variable, besides the three variables already in a Latin square design? To answer this
question, let us first number the internal blocks of the
usual Sudoku square of order n = 9 column-by-column,
as shown in Figure 6 (and in Figure 5).

Figure 7. Incidence matrix
for R u B (left) and B u C
(right).

796

The most striking feature of a Sudoku design follows
from its definition: Treatment (T ) is orthogonal to all
three variables: Row (R), Column (C) and Internal
Block (B); that is, the incidence matrices for T × R,
T × C, T × B are all J9. Of course, by the definition
of a square, the incidence matrix for R × C is already
J9 , yielding row–column orthogonality. The incidence
matrix for R × B and B × C are shown in Figure 7.

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

Hence, row and internal block variables are non-orthogonal; and column and internal 
block
variables, too are

4
non-orthogonal. Thus, of the 2 = 6 possible pairs of
variables, 4 pairs are orthogonal and 2 are not.
Returning to the example in Figure 5, we note that
Operators 1, 2, 3 are available on Days 1, 2, 3; Operators 4, 5, 6 on Days 4, 5, 6; and Operators 7, 8, 9 on Days
7, 8, 9. In spite of this restricted availability of the 9 operators, Machines are orthogonal to Suppliers, Days and
Operators. Why is orthogonality so desirable? Because
when orthogonality is achieved, it is known that comparison among the nine Machine Effects can be made
with maximum accuracy! This precisely is the statistical utility resulting from the combinatorial beauty of a
Sudoku design!
However, this statistical analysis of a Sudoku design [13]
has the following shortcomings: No operator worked
with raw materials from all 9 suppliers, and no operator worked on all 9 days. Consequently, operator–
supplier and operator–day pairs of variables were nonorthogonal. This went unnoticed in the analysis provided in [13]. Subsequently, this was properly accounted
for in [4]. Their analysis is quite intricate, but is not essential for this article.
In fact, if orthogonality between pairs of variables was
the only concern, the literature already contains a perfect solution to ensure such orthogonality: One can simply choose a pair of orthogonal Latin squares (OLS) (or
a Graeco–Latin squares or Euler squares), one representing Machine allocation and the other Operator assignment. Recall that two Latin squares are said to
be orthogonal if, when superimposed, they exhibit every ordered pair of numbers exactly once. There is a
well-developed mathematical theory for constructing a
complete set of OLS. In fact, a pair of OLS exist for all
orders n ≥ 3, but n = 6 [16]. Figure 8 displays a pair of

RESONANCE ¨ September 2015

797

GENERAL ¨ ARTICLE

Figure 8. An Orthogonal
Latin Square Design (OLSD)
of order 9.

9 × 9 OLS, forming an orthogonal Latin square design
(OLSD) of order 9, wherein the first entry represents
Machine and the second entry represents Operator.
Do mutually orthogonal Sudoku squares (OSS) (or Graeco Sudoku squares) exist? Indeed, they do [17]. Also,
see [18] for the construction of OSS of odd order. As an
illustration, the example in Figure 9 superimposes an
orthogonal (diagonal) Sudoku square to the orthogonal
(diagonal) Sudoku square of Figure 3.

Figure 9. An Orthogonal (diagonal) Sudoku Square Design (OSSD) of order 9.

798

Note, however, that in any OLSD or in any OSSD, every operator must be available to work on every day. In
contrast, for the design shown in Figure 5, each operator is required to work on only 3 days, which results in a
significant operational efficiency. Thus, when operator
availability is restricted, one cannot invoke an OLSD or
an OSSD. On the other hand, the Sudoku design of Figure 5 accommodates restrictions on the operator availability only at the cost of imposing non-orthogonality
for both operator–supplier and operator–day. But, as
already reasoned above, it is highly desirable that each

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

operator should work with raw materials from all 9 suppliers. Thus, even though we are willing to sacrifice
operator–day orthogonality in order to accommodate restricted availability of operators, we would like to preserve operator–supplier orthogonality. Is that possible?
Indeed, it is! We display one such Sudoku design in
Figure 10.

Figure 10 . A Sudoku design
with orthogonal incidence
structure for operator–supplier, when operators are
available as a team of three
in sets of three days.

In Figure 10, as in Figure 5, the incidence matrices for
S × D, S × M, D × M, O × M are all J9 . Additionally,
in Figure 10, the incidence matrix for S × O is also J9,
unlike that for R × B in Figure 5. But the incidence
matrix for O × D in Figure 10, as for B × C in Figure
5, is not J9. Thus, of the 6 possible pairs of variables
only one pair exhibits non-orthogonality.
The design in Figure 10 is an illustration of a special
sub-class of Sudoku designs called the cylindrical-shift
Sudoku designs. We do not discuss any statistical properties of such Sudoku designs in this article. Nor do
we discuss how to incorporate additional explanatory
variables orthogonal to the existing variables, or how
to allow other scenarios of operator availability such as
Operator i is available on days i, i + 1, i + 2 (modulo 9),
or on days i, i + 2, i + 4 (modulo 9), and still maintain
operator–supplier orthogonality. For such matters, we
refer interested readers to [19].

RESONANCE ¨ September 2015

799

GENERAL ¨ ARTICLE

4. Irregular Sudoku Puzzles
We conclude this article by displaying some irregular Sudoku puzzles that appeared in [20] (Figure 11). Here, ‘irregular’ means the bordered internal blocks are allowed
to be of any shape (not necessarily squares/rectangles)
formed by contiguous cells. Although not absolutely
necessary, the irregular-shaped (or squiggly) internal blocks usually exhibit, perhaps for aesthetic reasons, some
form of symmetry – lateral reflection, vertical reflection,
quarter-turn (90o rotation), or half-turn (180o rotation).
Figure 11. Some irregular
Sudoku puzzles.

800

Furthermore, although not required, in all of these irregular Sudoku puzzles, the cells in which values are already
given exhibit either a reflection or a rotation symmetry!

RESONANCE ¨September 2015

GENERAL ¨ ARTICLE

Glossary of Some Terms
Units: Single entities on which experimental conditions are applied and on which measurements are taken; as
for example, plots in an agricultural experiment.
Response variable: The main variable of primary interest in a statistical study; as for example, yield of a variety
of wheat grown in a plot.
Explanatory variables: Other variables that help us learn about the response variable once we can figure out
how these variables relate to one another; as for example, plot size, quantity of rainfall, amount of manure, etc.
Treatment: An experimental condition assigned to the units by the experimenter; as for example, wheat
varieties applied in experimental plots.
Levels: Distinct values of a variable; as for example, small, medium, large quantities of manure applied per
plot.
Statistical experimental design: Assignment of treatments to units in a planned manner accounting for
various constraints; as for example, planning an agricultural experiment involving experimental plots, varieties
of wheat, different levels of soil fertility, different amount of irrigated water etc. When planned well, an
experimental design can extract maximum information using as few units as are necessary.
Latin square: A p u p array with entries taken from {1, 2, ..., p} in such a way that every row (column) contains
each symbol exactly once. A Latin square experimental design uses the rows of a Latin square as the p classes
of explanatory variable A, its columns as the p classes of explanatory variable B, and the p symbols as the levels
of a treatment variable. The total number of units is only p2 (and not p3 as a complete experiment would
require).
Orthogonal Latin squares, or Graeco–Latin squares, or Euler squares: Two Latin squares which when
superimposed, exhibit every pair of ordered symbols formed from {1, 2, ..., p} exactly once. An OLS
experimental design assigns levels of two treatment variables to each of the p2 experimental units.

Acknowledgment
The authors thank the associate editors and two anonymous referees for their guidance in determining the right
focus for this article, and in suggesting several mathematical properties of Sudoku squares.
Suggested Reading
[1]

Daily Sudoku Puzzle, 4 Aug 2014. http://dailysudoku.com/sudoku/

[2]

Daily Sudoku Puzzle, 3 Aug 2014. http://dailysudoku.com/sudoku/
archive/2014/08/2014-08-3\_solution.shtml

RESONANCE ¨ September 2015

801

GENERAL ¨ ARTICLE
[3]

T Sander, Sudoku graphs are integral, The Electronic Journal of

[4]

B Felgenhauer and F Jarvis, Enumerating possible Sudoku grids,

[5]

E Russell and F Jarvis, Mathematics of Sudoku II. Unpublished, http:/
/www.afjarvis.staff.shef.ac.uk/sudoku/russell\_jarvis\ _spec2.pdf

[6]

A M Herzberg and M R Murty, Sudoku squares and chromatic
polynomials, Notices of the American Mathematical Society, Vol.54,
No.6, pp.708–717, 2007.

[7]

G McGuire, B Tugemann and G Civario, There is no 16-clue Sudoku:
Solving the Sudoku minimum number of clues problem, Unpublished,
http://arxiv.org/pdf/1201.0749v1.pdf

[8]

Gordon Royle, Minimum Sudoku, http://staffhome.ecm.uwa.edu.au/
~00013890/sudokumin.php

[9]

T Davis, The Mathematics of Sudoku, Unpublished, http://
www.geometer.org/mathcircles/sudoku.pdf

[10]

How to Solve Sudoku Puzzles, About.com Puzzles, http://
puzzles.about.com/library/sudoku/blsudoku\_tutorial01.htm

Combinatorics, Vol.16, No.25, 2009.
Unpublished, http://www.afjarvis.staff.shef.ac.uk/sudoku sudoku.pdf

[11] The Math Behind Sudoku, Cornell University Department of Mathematics, http://www.math.cornell.edu/~mec/Summer2009/Mahmood/
Intro.html
[12] S ammel and J Rothstein, The number of 9 u 9 Latin squares, Discrete
Math, Vol.11, pp.93–95, 1975.
[13]

J Subramani, and K N Ponnuswamy, Construction and analysis of
Sudoku designs, Model Assisted Statistics and Applications, Vol.4,
pp.287–301, 2009.

[14]

F A Saba and B K Sinha, SuDoKu as an experimental design – Beyond
the traditional Latin square design, Statistics and Applications, Vol.12,
(1 & 2), pp.15–20, 2014.

[15]

C R Rao, Linear statistical inference and its applications, Wiley, New
York, 1973.

[16]

Orthogonal Latin Squares, Encyclopedia of Mathematics, Springer
and the European Mathematical Society, http://www.encyclopediaofmath.org/index.php/Orthogonal\_Latin\ _squares

Indiana University-Purdue
University

[17]

S Golomb, Proposed Problem, American Mathematical Monthly,
Vol.25, No.3, 268, 2006.

Indianapolis, USA

[18]

J Subramani, Construction of Graeco Sudoku square designs of odd
order, Bonfring International Journal of Data Mining, Vol.2, pp.37–
41, 2012.

[19]

J Sarkar and B K Sinha, Variants of SuDoKu as useful experimental
designs, Statistics and Applications, Vol.12, (1 & 2), pp.35–60, 2014.

[20]

Daily Sudoku Puzzle, 1–4 Aug 2014, http://dailysudoku.com/sudoku/
archive/index.html

Address for Correspondence
Jyotirmoy Sarkar1
Bikas K Sinha2
1

Email: jsarkar@iupui.edu
2

Indian Statistical Institute
Kolkata
Email:

bikassinha1946@gmail.com

802

RESONANCE ¨September 2015