# The Structure of Reduced Sudoku Grids and the Sudoku Symmetry Group

Authors: Siân K. Jones; Stephanie Perkins; Paul A. Roach
Journal: International Journal of Combinatorics
 (2012)

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## Abstract

A Sudoku grid is a constrained Latin square. In this paper a reduced Sudoku grid is described, the properties of which differ, through necessity, from that of a reduced Latin square. The Sudoku symmetry group is presented and applied to determine a mathematical relationship between the number of reduced Sudoku grids and the total number of Sudoku grids for any size. This relationship simplifies the enumeration of Sudoku grids and an example of the use of this method is given.


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## Body

## 1. Introduction

A Sudoku grid,Sx,y, is a n×n array subdivided into n minigrids of size x×y  (where n=xy). Sx,y consists of y  bands, each composed of x horizontally consecutive minigrids, and x  stacks, each composed of y vertically consecutive minigrids. Each x×y minigrid possesses x subrows, or tiers and y subcolumns, or pillars. The values 1,…,n are contained within the array in such a way that each value occurs exactly once in every row, column, and minigrid. The discussion that follows requires analysis of the enumeration of Sudoku grids and therefore the following notation is required: Sx,y(n) is the number of ways of arranging the values in Sx,y.A Sudoku grid is a constrained Latin square. The calculation of the number of Latin squares, of some dimension, is greatly simplified by only counting those Latin squares which are inreduced form. A Latin square, or rectangle, is reduced if the values in the first row and column are in the natural order [1]. There are far fewer reduced Latin squares than Latin squares, and there exists a mathematical relationship allowing a direct calculation of the number of the latter from the number of the former. If the first row and first column of a Sudoku grid are in the natural order there is no longer a direct mathematical relationship between the number of these and the number of Sudoku grids. Therefore reduced Sudoku grids, defined in this paper, have different properties to reduced Latin squares.In [2–5] the number of reduced Latin squares (for sizes 8 to 11) is calculated. In a recent article, Stones [6] surveys some well-known and some more recent formulae for Latin rectangles, their usefulness, and means to obtain approximate numbers. If l(k,n) is the number of reduced Latin rectangles of size k×n then the total number of Latin rectangles, L(k,n), [1] is as follows:
(1.1)L(k,n)=n!(n-1)!l(k,n)(n-k)!.
A similar relationship is developed here between the total number of Sudoku grids and the number of reduced Sudoku grids. Such relationships have previously been given for “NRC-Sudoku” [7] and “2-Quasi-Magic Sudoku” [8] where the focus is the symmetry groups for these structures; symmetry groups have been defined for S3,2 [9], S3,3 [10], and the symmetry group of Sn,n is given in this paper.


## 2. The Sudoku Symmetry Group

The Sudoku symmetry group,𝔖, (containing symmetry operations applicable to Sx,y) consists of all homomorphisms of the structure of a Sudoku grid. An element of the symmetry group when applied to a Sudoku grid preserves the underlying structure of that grid (including the relationship between the values) while permuting the values contained within the grid. An element of the symmetry group α operating on a Sudoku grid s is written α(s). For the symmetry group, identity i is defined as i(s)=s. For any element α in the symmetry group, if α(s)=s then the Sudoku grid s is said to be “fixed” by the symmetry operation α. For two Sudoku grids s1 and s2, and two elements of the symmetry group α and β, if α(s1)=β(s2) then s1 and s2 are said to be isomorphic.Definition 2.1.
The symmetry group,𝔖, for a Sudoku grid is formed by the permutations in Table 1.Table 1
Permutation operations generating𝔖.
Permutation operation
Order
Permutation of the values (for example replacing all the values 1 in the entire grid with the value 2 and vice versa)
n
!
Permutation of bands
y
!
Permutation of stacks
x
!
Permutation of the columns within the stacks
y
!
x
Permutation of the rows within the bands
x
!
yBy Definition2.1 the Sudoku symmetry group 𝔖 consists of all permutation operations listed in Table 1 and any combination thereof. Each permutation operation is given in general and the order of each gives the total number of such operations. The cardinality of the Sudoku symmetry group is given by the product of the orders of each permutation operation:
(2.1)|𝔖|=n!x!y+1y!x+1.


## 3. Reductions

The properties of a reduced Sudoku grid are given in Definition3.1 and a direct mathematical relationship is given between the number of ways of arranging the values in a Sudoku grid and the number of ways of arranging the values in a reduced Sudoku grid, in Theorem 3.2. The reduced Sudoku properties are used in Section 4 to enable the simplification of the calculation of the number of ways of arranging the values in a Sudoku grid.Definition 3.1.
Areduced Sudoku grid,sx,y, is a Sudoku grid, Sx,y, having the following properties: (i)
the values inSx,y1,1 are in canonical form, [Sx,y1,1]i,j=(i-1)y+j;(ii)
for each minigridSx,y1,b, for b=2,…,x, the values in [Sx,y1,b]1,j for j=1,…,y are increasing;(iii)
for each minigridSx,ya,1, for a=2,…,y, the values in [Sx,ya,1]i,1 for i=1,…,x are increasing;(iv)
[Sx,y1,b]1,1<[Sx,y1,b+1]1,1 for b=2,…,x-1;(v)
[Sx,ya,1]1,1<[Sx,ya+1,1]1,1 for a=2,…,y-1.An example reduced Sudoku grid is given in Figure1(a), and an isomorphic Sudoku grid that can be formed from it (by permuting the values (1,9,2,5,8)(3,7,6)(4) and permuting the rows and columns) is given in Figure 1(b).An example Sudoku grid in reduced form and a sudoku grid isomorphic to it.
(a)
A reduced Sudoku grid
(b)
A Sudoku grid isomorphic to (a)Theorem 3.2.
Ifsx,y(n) is the number of reduced Sudoku grids of size n×n with minigrids of size x×y (where n=xy) then Sx,y(n) is given by the following:
(3.1)Sx,y(n)=(n-1)!x!yy!xsx,y(n).Proof.
Let a setX contain all reduced Sudoku grids. There exists an x,y∈𝔖 such that x(S)=y(T) for S,T∈X. Hence a count of the number of possible isomorphisms by applying the symmetry operations in 𝔖 will be larger than the number of distinct isomorphisms. The permutations in Table 2 contain all symmetry operations of 𝔖, and form the group ℝ, and is such that if p(S)=r(T) for S,T∈X then x=y and S=T for p,r∈ℝ.
The cardinality ofℝ is given by the product of the order of the permutation operations in Table 2, and thus gives the number of isomorphic Sudoku grids which may be formed from a given reduced Sudoku grid. Thus the total number of Sudoku grids may be calculated from the total number of reduced Sudoku grids using the following:
(3.2)Sx,y(n)=n!y!(x-1)x!(y-1)(x-1)!(y-1)!sx,y(n).
Equation (3.2) can be simplified, since n=xy, to give (3.1).Table 2
Permutation operations used to form isomorphic Sudoku grids.
Permutation operation
Order
Permutation of values1,…,n (equivalent to rearranging the values in Sx,y1,1)
n
!
Permutation of rows in bands2,…,x
(
y
!
)
(
x
-
1
)
Permutation of columns in stacks2,…,y
(
x
!
)
(
y
-
1
)
Permutation of bands2,…,x
(
x
-
1
)
!
Permutation of stacks2,…,y
(
y
-
1
)
!


## 4. Example of the Enumeration Technique

Let a4×4 Sudoku grid be in reduced form such that: the values in S2,21,1 are in canonical form, the values in [S2,21,2]1,j for j=1,2 are increasing; the values in [S2,22,1]i,1 for i=1,2 are increasing (note that the remaining two properties of reduced Sudoku grids are not needed for S2,2 since there are only two bands and two stacks). A partially filled reduced Sudoku grid is given in Figure 2 (containing only those values which are prearranged by the grid being in reduced form).Figure 2
A partially filled reducedS2,2.Consider the minigridS2,21,2; the values 1 and 2 must be contained in the second tier and there are two ways of arranging these values. Consider the minigrid S2,22,1; the values 1 and 3 must be contained in the second pillar and there are two ways of arranging these values. Four S2,2 grids are formed and are given in Figure 3. In Figures 3(a), 3(b), and 3(c) there is one way of arranging the values in S2,22,2 and in Figure 3(d) there are no valid ways of arranging the values in S2,22,2; therefore s2,2(4)=3.Four arrangements of the values in the partially filled reducedS2,2.
(a)
(b)
(c)
(d)The enumeration of the number ofS2,2 grids has been greatly simplified by only calculating the number of reduced Sudoku grids. Using (3.1) in Theorem 3.2 the total number of 4×4 grids can be calculated such that s2,2(4)=288.


## 5. Comparison of the Number of Known and Reduced Sudoku Grids

Exact values forSx,y(n) are known for x≤3 and y≤4. All results were calculated computationally and appeared around late 2005 on the Sudoku Players Forum [11], except for S3,3(9) which appeared in [12] in January 2006. All results have been confirmed computationally by other contributors to the forum. S2,2(4) is attributed to many different authors, S2,3(6), S2,5(10), and S2,6(12) are attributed to Pettersen [11]. The number S2,4(8) was calculated by Russel [11], S3,3(9) by Felgenhauer and Jarvis [12], and S3,4(12) by Pettersen and Silver [11]. Currently no other results are available in the academic or nonpeer-reviewed literature. Known numbers of Sudoku grids, Sx,y1,x(n), for some values of x and y are extended using Theorem 3.2 to produce a comparison of sx,y(n) and Sx,y(n); these are shown in Table 3.Table 3
Comparison ofSx,y(n) and sx,y(n).
x
,
y
S
x
,
y
(
n
)
s
x
,
y
(
n
)
2, 2
288
3
2, 3
28,200,960
1,632
2, 4
29,136,487,207,403,520
3,763,703,808
3, 3
6,670,903,752,021,072,936,960
56,738,340,804,608
5, 2
1,903,816,047,972,624,930,994,913,280,000
273,250,346,759,004,487,680The link between Latin squares and reduced Latin squares has been employed to great advantage, especially within the field of enumerative combinatorics. Here we have introduced a reduced Sudoku grid which likewise simplifies the enumeration of Sudoku grids. It can be seen from Table3 that there are far fewer reduced Sudoku grids than their regular counterparts and in Section 3 the mathematical link between them is demonstrated.



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*Source: 760310-2012-11-07.xml*