International Journal of Statistics and Applications 2016, 6(5): 325-327
DOI: 10.5923/j.statistics.20160605.07
Construction and Analysis of Partially Balanced
Sudoku Design of Prime Order
A. Danbaba*, N. S. Dauran
Department of Mathematics, Usmanu Danfodiyo University, Sokoto, Nigeria
Abstract Sudoku squares have been widely used to design an experiment where each treatment occurs exactly once in
each row, column or sub-block. For some experiments, the size of row (or column or sub-block) may be larger than the
number of treatments. Since each treatment appears only once in each row (column or sub-block) with an additional empty
cell such designs are partially balanced Sudoku designs (PBSD) with NP-complete structures. This paper proposed methods
for constructing PBSD of prime order of treatments by a modified Kronecker product and swap of matrix row (or column) in
cyclic order. In addition, linear model and procedure for the analysis of data for PBSD are proposed.
Keywords Sudoku design, Partial Sudoku, NP-complete, Kronecker product, Raw and column swap
1. Introduction
A Sudoku design is a block matrix filled with different
Latin letters, such that each letter appears in a row (column
or sub-block) only once. Hui-Dong and Ru-Gen (2008)
stated that all non-prime number k can construct a Sudoku
square and some of them can constitute more than one, and
prime number k cannot, where k is the number of treatments
(or rows or columns) of the square. Subramani and
Ponnuswamy (2009) discussed the construction of Sudoku
designs of order k = m2 only. They proposed linear models
for analyzing the data obtained from their design, and
applied the designs to agricultural experiments. Recently, It
has been shown that Sudoku design may be partial or
incomplete (Béjar et al. 2012; Mahdian, and Mahmoodian
2015). In addition, Donovan et al. (2015) and Kumar et al.
(2015) studied the Sudoku based space filling designs.
A partial Sudoku design is a partially filled block matrix,
with some empty cells, which also satisfies that each Latin
letter appears only once in row (column or sub-block), see
Mahdian, and Mahmoodian (2015). This partial Sudoku
design has been shown to be NP-complete for the particular
case of square sub-blocks (n rows and n columns in each
sub-block), see Kanaana and Ravikumar (2010). It was
reported by Béjar et al. (2012) that, even when sub-blocks
are not square the completion problem is also NP-complete.
In general, it is an NP-complete problem to determine if a
partial Sudoku square is completable (Colbourn 1984;
Mahdian, and Mahmoodian 2015).
* Corresponding author:
[email protected] (A. Danbaba)
Published online at http://journal.sapub.org/statistics
Copyright © 2016 Scientific & Academic Publishing. All Rights Reserved
In this paper, methods on how to construct and analyze
datawith partial Sudoku design when sub-blocks are not
square for which the number (k) of treatments (Latin letters)
in each row (column or sub-block) is prime are proposed.
2. Construction of Partially Balanced
Sudoku Design
Consider a sub-block (D11) of n Latin letters in a block
matrix D. To construct a Sudoku design is to simply fix D11
in the first row and first column of D, then perform raw swap
of D11 in cyclic order to obtain D12 in the first raw and second
column of D and so on until all the Latin letters occur once in
each row of the first row-block of D. Then perform column
swap of D11 in cyclic order to obtain D21 in the second row
and first column, and then perform row swap of D21 in cyclic
order to obtain D22inthe second row and second column of D
and so on until all the Latin letters appear once in each row of
the second row-block of D. Repeat this procedure until each
Latin letter appears once in each column of the
column-blocks of D. Then D is a Sudoku design.
Alternatively, performing the Kronecker product of matrix
of one’s (J) and D11, we can form the bock matrix D with
sub-block matrices formed by raw (or column) swap in a
cyclic order of D11.
Now, we presented the construction of Sudoku designs
where sub-blocks are not square and k is a prime number. In
this case D11 is an initial sub-block of size greater than the
number of treatments. We consider only k = 5 and 7. Suppose
k = 5, and let a, b, c, d, and e be the treatments such that the
sub-block matrix.
A. Danbaba et al.:
326
Construction and Analysis of Partially Balanced Sudoku Design of Prime Order
where yijrl is the response on the ith treatment in the jth row,
a b


D11 =  c d  .
e



rth column and lth sub-block, μ is the overall mean, α i is the
ith treatment effect, β j is the jth row effect, γ k is the rth
Then performing row (or column) swap of D11, we
obtained:
a b   c d   e


 
 

 c d   e
  a b 
  a b   c d 
  e
 
 


D=
e 
 b a d c  

 
 

e   b a
d c  

e   b a   d c  

H α : all α i are equal
H β : all β j are equal
H γ : all γ r are equal
Hθ : all θl are equal
All the null hypotheses ( H α , H β , H γ , Hθ ) of main-effect
a b


c
 and J 1 1 1
=
D11 =


d e
 1 1 1


 f g
Then perform the Kronecker product of j and D11 together
with row (or column) swap of D11, we have:
1 1 1 1
 ⊗
1 1 1 1
=
D 
  d e   f g 
b  c
 
 

 
  d e   f g   a b 

e  f g  a b  c

 





  d e 
g   a b   c
 

a

c
d

f 


e
g

b
c

d
f

a 
e

g
b



d

f
a

c 
g

b


e

the error term such that ε ijrl  iiN (0, σ 2 ) .
Hypotheses of interest are as follows:
The design is balanced since each row, column or
sub-block contained the five treatments (atoe). It is partial
Sudoku (incomplete Sudoku) because each sub-block has an
empty cell. Note that for prime number k greater than 2,
Sudoku design can be constructed for which sub-blocks are
not square. To illustrate the use of Kronecker product, let k =
7 and a, b, c, d, e, f, g be treatments such that:
 a

 c
 d

 f
D11 =  
 b


 e
 
 g

column effect and θl is the lth sub-block effect, ε ijrl is

f 

a 
c 

d  
are rejected at a level of significant if
F =
MS x
where x= 1, 2, 3, 4.
: Fdf
x ,m ( m−3)
MS E
Table 1. ANOVA table for experimental data from a Sudoku square design
with non-square blocks
Source
df
Treatments
k–1
SS1
=
Rows
k
=
SS 2
∑ k + 1 − k (k + 1)
Columns
k
SS3
=
−
∑
=1 k + 1 k ( k + 1)
SS 4
=
−
∑
i =1 k + 1 k ( k + 1)
Blocks
k
Error
k(k– 3)
Total
SS
MS
yi2•••
y2
− ••••
k
k (k + 1)
k
∑
i =1
k
y•2 j ••
2
y••••
y••2 r •
2
y••••
2
y•••
l
2
y••••
i =1
k +1
i
k +1
SSE =SST – SS1 – SS2 – SS3 – SS4
SST
=
k(k+1)–1
k +1 k +1
∑
∑y
j =1
k
2
( ij ) rl
−
MS1
MS2
MS3
MS4
MSE
2
y••••
k (k + 1)
3. Analysis
4. Conclusions
The following linear model with treatments, rows,
columns and sub-block is proposed:
In this paper, two methods of constructing partially
balanced Sudoku designs were developed under non-square
sub-blocks each containing prime number of treatments. The
first method developed uses permutations of row (or column)
in cyclic order to construct the design while the second
method uses a modified Kronecker product of matrices in the
construction. A linear model for the analysis of data for the
designs is also developed.
yijrl
i = 1, 2,..., k
 j 1, 2,..., k + 1
=

= µ + α i + β j + γ r + θl + ε ijrl 
=
r 1, 2,..., k + 1
l 1, 2,..., k + 1
=
International Journal of Statistics and Applications 2016, 6(5): 325-327
327
[4]
Hui-Dong, M. and Ru-Gen, X. (2008). Sudoku Square — a
New Design in Field Experiment, ActaAgron Sin, 34(9),
1489–1493.
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