A KNOWLEDGE-RICH APPROACH TO THE RAPID

ENUMERATION OF QUASI-MAGIC SUDOKU SEARCH SPACES

P. A. Roach, S. K. Jones, S. Perkins

Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, U.K.

I. J. Grimstead

Cardiff School of Computer Science, Cardiff University

Queen’s Buildings, 5 The Parade, Roath, Cardiff, CF24 3AA, U.K.

Keywords:

Search, Constraints, Quasi-Magic Sudoku, Coding Theory.

Abstract:

The popular logic puzzle, Sudoku, consists of placing the digits 1, . .. , 9 into a 9 × 9 grid, such that each digit

appears only once in each row, column, and subdivided ‘mini-grid’ of size 3 × 3. Uniqueness of solution

of a puzzle is ensured by the positioning of a number of given values. Quasi-Magic Sudoku adds the further

constraint that within each mini-grid, every row, column and diagonal must sum to 15 ±∆, where ∆ is chosen to

take a value between 2 and 8. Recently Sudoku has been shown to have potential for the generation of erasure

correction codes. The additional quasi-magic constraint results in far fewer given values being required to

ensure uniqueness of solution, raising the prospect of improved usefulness in code generation. Recent work

has highlighted useful domain knowledge concerning cell interrelationships in Quasi-Magic Sudoku for the

case ∆ = 2, providing pruning conditions to reduce the size of search space that need be examined to ensure

uniqueness of solution. This paper examines the usefulness of the identiﬁed rich knowledge in restricting

search space size. The knowledge is implemented as pruning conditions in a backtracking implementation of

a Quasi-Magic Sudoku solver, with a further cell ordering heuristic. Analysis of the improvement in processing

time, and thereby of the potential usefulness of Quasi-Magic Sudoku for code generation, is provided.

1 INTRODUCTION

The popular number-based logic puzzle, Sudoku,

consists of a 9 × 9 grid which is further subdivided

into ‘mini-grids’ of size 3 × 3. The values 1, . . . , 9

are to be placed into the grid, such that each digit ap-

pears only once in each row, column, and mini-grid.

The puzzle has given rise to many variants of differ-

ent sizes and border constraints. Quasi-Magic Sudoku

adds the further constraint that within each mini-grid,

every row, column and diagonal must sum to 15 ± ∆,

where ∆ is chosen to take a value between 2 and 8.

In both the Sudoku and Quasi-Magic Sudoku

cases of the puzzle, an initial puzzle state contains

a partial assignment of values to the cells. These

‘given’ values may not be moved, and are chosen to

ensure that solvers may arrive only at a single valid

grid arrangement - the unique solution. A minimum

number of givens should be offered, such that no

given is itself a ‘logical consequence’ (Simonis, 2005)

of the other givens (i.e. no redundant clues are pro-

vided). The smallest known number of givens for a

well-formed puzzle is 17 (Royle, 2006), but no gen-

eral means is known for proving the minimum num-

ber necessary (Bartlett and Langville, 2006). The

missing values are entered so as to satisfy the con-

straints of the puzzle, either by hand through the ap-

plication of logic, or by some automated solver. A

complexity rating of a puzzle (such as ‘easy’, ‘hard’

or ‘diabolical’) is a subjective measure of the time re-

quired for its manual solution. It is worth noting that

this rating does not have a simple relationship with

the number of givens - the quantity of puzzle givens

is generally less important as a measure of their value

as ‘hints’ than is their positioning in the grid (Jones

et al., 2007).

Through its relationships with Latin Squares and

other combinatorial structures, Sudoku is strongly

linked to many real-world applications, including

conﬂict free wavelength routing in wide band opti-

cal networks, and timetabling and experimental de-

sign (Gomes and Shmoys, 2002). More importantly

246

Roach P., Jones S., Perkins S. and Grimstead I. (2009).

A KNOWLEDGE-RICH APPROACH TO THE RAPID ENUMERATION OF QUASI-MAGIC SUDOKU SEARCH SPACES.

In Proceedings of the International Conference on Agents and Artiﬁcial Intelligence, pages 246-254

DOI: 10.5220/0001659502460254

 SciTePress

here, Sudoku has also been shown to be useful for

the construction of erasure correcting codes (Soedar-

mandji and McEliece, 2007), where the three inherent

constraints of the Sudoku grid are used to form gen-

eralised parity check operations. In (Soedarmandji

and McEliece, 2007) it was demonstrated that Su-

doku codes can be decoded iteratively in a manner

similar to that used for Low Density Parity Check

codes. Sudoku codes can also be mapped onto graph

codes, which are currently attracting huge interest in

the coding theory literature as they perform at a level

close to the ‘gold standard’ of the Shannon capacity

(McWilliams and Sloane, 1977). Quasi-Magic Su-

doku offers the prospect of a similar approach to era-

sure code construction, but with the potential for grid

reconstruction from a smaller set of givens. This re-

duction in necessary givens results from the addition

of the quasi-magic constraint as an extra generalized

parity check operation. Well-formed Quasi-Magic

Sudoku puzzles have been produced with as few as

four givens (Forbes, 2007a).

Trivially, all Quasi-Magic Sudoku solution grids,

for a speciﬁed value of ∆, form a subset of all Sudoku

solution grids, and it is for this reason that they are the

focus of this paper. The authors contend that efﬁcient

Sudoku solvers may be implemented by exploiting

rich interrelationships between cell values. Recent

work has identiﬁed such interrelationships for Quasi-

Magic Sudoku for the case ∆ = 2 (Forbes, 2007a),

(Jones et al., 2008). This rich property domain infor-

mation enables the construction of pruning conditions

that greatly reduce the search space to be examined

for a solution to a puzzle.

Ultimately, the authors wish to establish that a par-

tial received message may be used to reconstruct the

original message - and that it may be done reliably,

and in a reasonable length of time. In the same way

as for Sudoku codes (Soedarmandji and McEliece,

2007), each q×q Quasi-Magic Sudoku puzzle (a stan-

dard grid has q = 9) can be viewed as a q-ary code-

word of length q

with erasures. Successful unique

completion of the grid is equivalent to unique decod-

ing of the erasure code. To ensure that the correct

grid has been reconstructed, it is necessary to estab-

lish the uniqueness of the result obtained, which we

achieve in this paper through full enumeration of the

search space of a puzzle. As a ﬁrst step towards this

aim, we can consider the process of reconstruction of

a puzzle grid - reliably in a rapid processing time - as

the solution of a puzzle from its (small) set of givens.

Establishing that the use of domain-speciﬁc informa-

tion can suitably reduce the size of the problem search

space that needs to be enumerated is viewed here as

important in this ﬁrst step.

This paper highlights the domain-rich cell value

interrelationships in Quasi-Magic Sudoku for the case

∆ = 2 in Section 3. It then describes a fast backtrack-

ing implementation of a Quasi-Magic Sudoku solver

that fully enumerates the search space in Section 4.

The rich knowledge concerning the interrelationships

of cell values in Quasi-Magic Sudoku is incorporated

into the solver, providing pruning conditions that re-

duce search space size. A further heuristic for rapid

search through cell ordering is proposed. An analy-

sis of the improvement in processing time is provided

in Section 5, along with a discussion of the potential

of Quasi-Magic Sudoku for erasure correction code

construction.

2 LITERATURE SURVEY

At present, only a very limited amount of work has

been published on Quasi-Magic Sudoku. Some re-

sults concerning the properties of the puzzles have

been reported (Forbes, 2007a), and some of these

properties have been proved mathematically (Jones

et al., 2008). These are detailed in Section 3. No work

has yet been published speciﬁcally on the automation

of solution of Quasi-Magic Sudoku, and so this sur-

vey will focus on those approaches taken to construct-

ing Sudoku solvers that are most relevant here.

Heuristic search optimisation algorithms directly

exploit features of a problem domain in order to re-

duce time spent examining a search space (the space

of all possible solutions to a given problem) (Rich and

Knight, 1991). Some or all of the problem constraints

are incorporated into an optimisation objective func-

tion that guides the search process. This approach

relies on being able to construct an objective func-

tion that reliably distinguishes between states that are

nearer or further away from a goal state (the solution).

This has been reported by the authors to be difﬁcult to

achieve in Sudoku (Jones et al., 2007), as the limited

amount of useful domain information that can be in-

corporated into the objective function tends to result

in many states mapping to the same score.

Recently, meta-heuristic approaches have been

applied to the solution of Sudoku. These include geo-

metric particle swarm optimisation (Moraglio and To-

gelius, 2007) and genetic algorithms (Moraglio et al.,

2006). It might seem that meta-heuristics are ap-

propriate, in particular those that employ pools of

solutions and possibly means of mutating solutions,

to avoid local maxima and plateaus in the objective

function (Jones et al., 2007). However, these elabo-

rate schemes are probably not justiﬁed and are often

extremely inefﬁcient. It has been demonstrated that

A KNOWLEDGE-RICH APPROACH TO THE RAPID ENUMERATION OF QUASI-MAGIC SUDOKU SEARCH

SPACES

247

even the combination of a sensible choice of the ini-

tial puzzle conﬁguration and simple heuristics, imple-

mented in a standard local search optimisation algo-

rithm (modiﬁed steepest ascent hill-climbing) is suf-

ﬁcient to ensure the reliable and reasonably efﬁcient

solution of Sudoku puzzles of different complexity

ratings (Jones et al., 2007). Approaches to solving

Sudoku puzzles that can employ domain information

to greatly prune the search space that need be consid-

ered in locating the solution, while simple, may prove

effective. That is the approach taken in this paper,

in analysing the potential usefulness of Quasi-Magic

Sudoku for code generation.

3 PROBLEM SIZE AND CELL

INTERRELATIONSHIPS

For Quasi-Magic Sudoku, the additional constraint

that within each mini-grid, every row, column and di-

agonal must sum to 15 ± ∆ is applied. For the case

∆ = 0, it may trivially be shown that there are no

valid grids (as the value 5 must lie in the centre of

a mini-grid satisfying this property, but 5 cannot lie in

the centre of every mini-grid (Forbes, 2007a)). For

the case ∆ = 1, a similar result holds for the posi-

tioning of the values 1 and 9 (Forbes, 2007a). For

∆ = 9, no additional constraints are being imposed,

leading to a standard Sudoku grid (as every row, col-

umn and diagonal in a Sudoku mini-grid will nec-

essarily sum to a value in the range 6, . . . , 24); the

number of valid grid arrangements is known to be

6, 670, 903, 752, 021, 072, 936, 960 (Felgenhauer and

Jarvis, 2006).

The cases of ∆ in the range 2, . . . , 8 are of greater

interest here, but only for the case ∆ = 2 is it cur-

rently known how many valid grids are possible. This

number, 248, 832, is reported as a new result here,

determined through enumeration using the solver de-

scribed in Section 4, and proven mathematically in

(Jones et al., 2008). No work has yet been published

on other cases, hence we pursue the case ∆ = 2 in this

paper. Even in this heavily constrained case, these are

many possible valid grids. Hence, even with givens

added to an empty grid, a typical search space, for

example one arising from a local search optimisation

approach to solving grids, is deceptively large.

The imposition of the additional Quasi-Magic Su-

doku constraint leads to interrelationships between

cell values that are speciﬁc to the value of ∆. These in-

terrelationships may be used to identify arrangements

of cell values that are not possible in a valid grid, lead-

ing to sets of rules that may be employed in pruning a

search space of possible grid solutions for any puzzle.

In order to explain the pruning rules for the case

∆ = 2, we introduce here some additional terminol-

ogy for Sudoku grids: mini-grids are organised into

bands (horizontally) and stacks (vertically). Hence,

each Sudoku grid has 3 bands of 3 mini-grids, and

3 stacks of 3 mini-grids. With this terminology, the

quasi-magic pruning rules can now be written as be-

low. The ﬁrst nine rules are derived and reworded

from results previously reported in (Forbes, 2007a),

and proved in (Jones et al., 2008) which also adds the

tenth rule as an extension to those results:

1. Only 3, 4, 5, 6 or 7 can be in the centre of any

mini-grid. (A 1, 2, 8 or 9 would violate the quasi-

magic constraint for one or more rows, columns

and diagonals within the mini-grid.)

2. The value 5 can only be placed in the centre cell,

or in a corner cell, of any mini-grid.

3. Every band and stack must contain exactly one

mini-grid with a 5 in its centre cell (and exactly

3 mini-grids have 5 in the centre cell within the

entire grid).

4. At most one mini-grid will have 3 in its centre

cell; the same applies for the value 7.

5. If there is a mini-grid with centre 3 and a mini-

grid with centre 7, then those mini-grids must be

either in the same stack or the same band.

6. The values 6 and 7 can not form mini-grid centres

in the same stack or band.

7. The values 3 and 4 can not form mini-grid centres

in the same stack or band.

8. In any mini-grid, the values 1 and 2 can not lie in

the same row, column or diagonal.

9. In any mini-grid, the values 8 and 9 can not lie in

the same row, column or diagonal.

10. If all mini-grid centres are 4, 5 and 6 (i.e. there are

no 3 or 7 centres) then the values 4 and 6 can only

lie in centre and corner cells in any mini-grids.

The givens rule out some possible placements, re-

ducing the size of the search space. (Note that each

given should be chosen such that it is neither a logi-

cal consequence of any other given, nor of the quasi-

magic rules; hence their consequence in ruling out

possible placements is relatively small, especially as

there are fewer of them than would be the case in Su-

doku.) The number of total possible combinations of

remaining values, and therefore the number of dis-

tinct states in the search space (an indicator of search

space size), is reduced to the number of permutations

of non-given values within their respective mini-grids

ICAART 2009 - International Conference on Agents and Artificial Intelligence

248

Figure 1: A Quasi-Magic Sudoku Grid (Forbes, 2007b).

(Jones et al., 2007). This is calculated as

∏

i=1

∏

j=1

i j

! (1)

where n

i j

is the number of unassigned cells in the

mini-grid at band i (i = 1, . . . , 3), stack j ( j = 1, . . . , 3).

For the example Sudoku grid shown in Figure 1, with

8 givens, this number is 8! × 9! × 8! × 8! × 7! × 8! ×

8! × 9! × 8! ≈ 2.851 × 10

4 THE SOLVER

The Quasi-Magic Sudoku puzzle is formulated here

as a state-based problem, requiring deﬁnitions for a

cell, a grid of such cells (the state), the means of mod-

ifying the state and pruning conditions to reduce the

size of the search space. This section describes these

components, explaining how the pruning conditions

of Section 3 are incorporated. The use of a heuristic,

that of ordering the cells to be ﬁlled according to how

many valid placements of values there currently are

for each cell, is also described. This encourages more

rapid reduction of the search space.

4.1 Deﬁnition of a State

A cell is deﬁned in two parts:

• a ﬂag vector, denoting which values may currently

still be assigned to that cell - the candidate values;

• a ‘just ﬁxed’ ﬂag, denoting that the content value

of the cell has recently been ﬁxed.

Initially, each cell has 9 candidate values, and its

value has not been ﬁxed. The ﬂag vector indicates

the candidate values through the use of 9 consecu-

tive bits - each bit representing a different value. Ini-

tially, all bits are set to 1. A value may be designated

as not assignable to a cell (for example because the

value has already been assigned to another cell in the

same row, column or mini-grid) by setting the bit to

zero. This process of masking bits removes a candi-

date value from the cell. Once a value has been as-

signed to a cell, the ‘just ﬁxed’ ﬂag of the cell is set

as a cue to consider the consequences of assignment.

We note that this approach mimics a technique that

may be employed by human solvers, in marking up

candidate numerals in the non-ﬁxed cells.

A puzzle state is simply a grid of 81 cells, and is

used to record the cells currently ﬁxed and the candi-

date values for all remaining non-ﬁxed cells.

Another important consideration in the state def-

inition is the formation of valid rows, columns and

diagonals in the mini-grids of a Quasi-Magic Sudoku

grid. A sequence of three values in a row, column or

diagonal of a mini-grid will be referred to as a triple; a

triple that conforms to the quasi-magic sum constraint

15 ± ∆ (∆ = 2) will be referred to as a qm-triple.

4.2 Search Technique

The search approach employs a backtracking al-

gorithm to perform a depth-ﬁrst search (Rich and

Knight, 1991) of a puzzle search space. At each

stage, it attempts to assign a value to the next cell

to be considered, by selecting the numerically lowest

of the remaining candidate values for that cell. This

assignment may have far-reaching consequences for

the grid, and it is necessary to propagate these conse-

quences throughout the grid (Section 4.3).

The quasi-magic pruning rules are split into two

groups, one of which is implemented whenever a cell

is assigned a value (Section 4.3); the other is imple-

mented in a pre-processing stage (Section 4.4). Two

approaches are taken to ordering the assignment of

values to cells (Section 4.5). Finally, the backtracking

algorithm itself is detailed in Section 4.6.

4.3 Propagating the Consequences of a

Cell Assignment

During the process of solving a puzzle, cells in the

grid will be ﬁxed to have a speciﬁc value. This will

happen either directly, as an attempted step in the so-

lution of a puzzle, or indirectly, as a result of the ﬂag

vector of a cell being reduced to a single candidate so-

lution. On each occasion that an assignment is made,

it is necessary to propagate the consequences of the

assignment throughout the grid. This involves ﬁrstly

removing the assigned value as a candidate from all

other cells in the same row, the same column, and the

same mini-grid. This is achieved by masking the ap-

propriate bit from the ﬂag vectors of all such asso-

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249

Algorithm 1 : Propagation Algorithm.

Grid is the matrix of cells

Current cell is the cell to which an assignment has

been made

Current value is the value assigned to Current cell

Valid grid is a Boolean ﬂag, initially set to TRUE

for each cell, Cell c, in the same row, column, or

mini-grid as Current cell do

Mask Current value from Cell c

if Cell c is reduced to now having a single can-

didate value then

set ‘just ﬁxed’ ﬂag of cell

end if

end for

if Grid contains a cell having no candidate values

then

set Valid grid to FALSE and terminate PROPA-

GATION

end if

for each row, column and diagonal in which Cur-

rent cell forms a triple do

if the triple is completed, but is not a qm-triple

then

set Valid grid to FALSE and terminate PROP-

AGATION

else if the triple has one unassigned cell then

Check the ﬂag vector of the unassigned cell

for candidate values

if the triple can not become a qm-triple then

set Valid grid to FALSE and terminate

PROPAGATION

end if

end for

for each cell, Cell c, in Grid having ‘just ﬁxed’ ﬂag

set do

Clear ‘just ﬁxed’ ﬂag of cell

Call PROPAGATION recursively

for each rule, Rule r, of the pruning rules 3 to 10

that can apply to Cell c do

if Rule

r is violated by Cell c then

set Valid grid to FALSE and terminate

PROPAGATION

end if

end for

ciated cells. However, this process may have other

consequences for the grid:

• an updated cell may be reduced to having just one

remaining candidate;

• an updated cell may be reduced to having no re-

maining candidates;

• a qm-triple constraint may be violated;

• a quasi-magic pruning rule may be violated.

The assignment of a value to a cell is indicated

by the ‘just ﬁxed’ ﬂag of the cell becoming set, trig-

gering a call to the Propagation Algorithm outlined

above. This algorithm returns a Boolean value indi-

cating whether an assignment is valid, but it also al-

ters the current grid, restricting ﬂag vectors and ﬁxing

cells that have indirectly been limited to a single can-

didate value. A grid is rejected if any cell is reduced to

having no remaining candidate values, as this makes

the grid impossible to complete.

A grid is rejected if a completed triple is deﬁnitely

not a qm-triple. It is also rejected if a triple having one

non-ﬁxed cell can not become a qm-triple given its

remaining candidate values. (Note that the algorithm

does not check whether a triple in which two cells

are currently not ﬁxed may still become a qm-triple;

it was deemed that this check would seldom result in

rejection and would cost more in processing time to

execute than would be saved in search space pruning.)

For every cell now having the ‘just ﬁxed’ ﬂag

set, the algorithm is called recursively to propagate

its consequences. The ‘just ﬁxed’ ﬂag of the current

cell is cleared to avoid unnecessary recursion. Any

pruning rule that might be violated by the ﬁxed cell

is checked. (Recall from Section 3 that the pruning

rules relate to speciﬁc positions in a mini-grid and so,

for example, a cell assignment at the corner of a mini-

grid could not possibly violate rule 6.)

4.4 Pre-processing of the Grid

Algorithm 2 : Pre-processing Algorithm.

for each given value do

call PROPAGATION

end for

Apply pruning rules 1 and 2

An empty grid is transformed to a puzzle state by

the addition of the given values. The placement of

a given value is treated simply as the assignment of

a cell to a speciﬁc value, masking all other candidate

values from the cell’s ﬂag vector. The setting of the

cell’s ‘just ﬁxed’ ﬂag causes the Propagation Algo-

rithm to be called, determining the consequences of

the assignment. At this stage, it is also useful to apply

quasi-magic pruning rules 1 and 2, restricting the can-

didate values of the mini-grid centres and the avail-

able positions for the valid placement of the value 5.

The Pre-processing Algorithm, to prune the search

space to take into account the givens and ﬁrst two

pruning rules, is as above.

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250

4.5 Cell Ordering

A further dynamic pruning of the search space is im-

plemented through an ordering of the assignments to

cells. Rather than considering cells in some consec-

utive order in terms of their physical position in the

grid, they could instead be ordered according to how

few remaining candidate values they possess. The

main tool in dynamically pruning the search space

is the propagation of the consequences of assigning

a value to a cell throughout the associated row, col-

umn and mini-grid; this effectively prunes up to 20

fruitless branches. The cell ordering heuristic follows

the belief that by favouring assignment to cells hav-

ing fewest candidates, this process of pruning through

propagation will be enacted more rapidly. It is impor-

tant to note that this heuristic does not cause useful

portions of the search space to be overlooked.

A comparison of solution times is made in Section

5 between row-by-row ordering (selecting all cells

across a row before moving to the next neighbouring

row) and this fewest-candidate ordering. We note that

this cell ordering would also seem to mimic a typical

solution strategy employed in manual solution - that

of scanning the grid for cells in which candidate val-

ues may be eliminated.

4.6 Backtracking Algorithm

The algorithm performs a full enumeration of that part

of the search space that has not been pruned by the

Propagation Algorithm. This means that the search

procedure does not stop when the solution is found,

but continues until all possible remaining states have

been enumerated, conﬁrming the uniqueness of solu-

tion (where appropriate). This is particularly useful

for the intended application of Quasi-Magic Sudoku

to the construction of erasure codes, for which it is

important to be certain that every grid can be recon-

structed from a partial grid in a reasonable length of

time. Hence the assurance that the resulting pruned

search space can rapidly be fully enumerated from a

partial grid of givens would be a useful ﬁrst step in

establishing the general applicability of Quasi-Magic

Sudoku. The backtracking algorithm is as below.

The iteration count, a global counter for the num-

ber of attempted direct assignments of values to cells,

is used in Section 5 as one measure of algorithm per-

formance. Note that many cells are ﬁxed indirectly,

as a result of the propagation algorithm. This will be-

come clearer in Section 5. The recursive call takes the

search process further down the current branch of the

search space (Rich and Knight, 1991).

Algorithm 3 : Backtracking Algorithm.

Grid is the matrix of cells

Set Current cell to be the next cell to be ﬁxed (ac-

cording either to row-by-row or fewest-candidate

ordering)

if there are no cells then

note Grid as a solution

else

for each candidate value in the ﬂag vector of

Current cell do

Increment the Iteration Count

Current value is the next candidate value of

Current cell

Create a local copy of the grid,

Grid local copy.

In Grid local copy, assign Current cell the

Current value

Call PROPAGATION

Call BACKTRACKING (recursively) with the

modiﬁed Grid local copy

end for

end if

5 RESULTS AND EVALUATION

To demonstrate the speed of the full enumeration of

the search space, a relatively low-speciﬁcation ma-

chine was chosen for all tests - an HP-Compaq tc1100

Pentium-m 1GHz portable tablet PC, with 1.5GB

RAM. The algorithm was implemented in Sun’s Java

1.6.0 5 run-time environment; the code was run in de-

bug mode, and was not optimised in any way (such as

obfuscation to reduce variable name lengths).

Quasi-Magic Sudoku puzzles are not commonly

published, but a test set of 180 puzzles was assembled

from (Forbes, 2007a) and (Forbes, 2007b). In Su-

doku, the relationship between the number of givens

and puzzle complexity (measured in accordance with

how long human solvers take to solve them) is not

simple (Jones et al., 2007). As it might reasonably be

assumed that this holds also for Quasi-Magic Sudoku

puzzles, and as no consistent measure of puzzle com-

plexity was generally available for the puzzles of the

test set, the results are categorised purely in terms of

the numbers of givens. The largest number of givens

was 18, and the smallest 4. (It is worth recalling that

although for Sudoku, the smallest reported number of

givens is 17 (Royle, 2006), the additional quasi-magic

rules enable uniqueness of solution to be guaranteed

with far fewer givens.)

The authors know of no published work on the

automated solution of Quasi-Magic Sudoku puzzles.

A comparison is presented here of four different ap-

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251

Table 1: Summary of results for all puzzles by solution method: numbers of iterations and processing times (s) for full

enumeration of non-pruned search space.

No q-m No q-m With q-m With q-m

pruning rules pruning rules pruning rules pruning rules

row-by-row fewest-candidate row-by-row fewest-candidate

average iterations 71097 30889 24457 2055

maximum iterations 755373 1172274 214060 51594

minimum iterations 105 28 103 4

median iterations 15963 1608 4658 226

average time (s) 1.6924 0.7984 0.5927 0.0519

maximum time (s) 19.0022 30.6051 5.4582 1.2986

average time per

iteration (ms) 0.0238 0.0258 0.0242 0.0252

proaches to solving the problem, to analyse the use-

fulness of the quasi-magic pruning rules of Section 3

and the fewest-candidate ordering of cells proposed in

Section 4.5. The latter is compared with row-by-row

ordering. All approaches employ checking of whether

mini-grid triples are qm-triples (Section 4.1). The ap-

proaches are:

• no quasi-magic pruning rules, row-by-row cell or-

dering;

• no quasi-magic pruning rules, fewest-candidate

cell ordering;

• quasi-magic pruning rules, row-by-row cell order-

ing;

• quasi-magic pruning rules, fewest-candidate cell

ordering.

Runs of the algorithm are analysed here as to the

time taken (in seconds, to 4 decimal places), and the

number of iterations (to the nearest integer) which had

to be performed to complete each puzzle (as deﬁned

in Section 4.5). As the backtracking algorithm per-

forms a full enumeration of that part of the search

space that has not been pruned by the pruning con-

ditions and qm-triple checks, the results presented

here are not skewed by puzzles for which the solution

is ‘conveniently’ positioned within the search space.

The results obtained are therefore an accurate mea-

sure of the relative sizes of the non-pruned search

spaces of the puzzles, and not just a measure of how

rapidly a solution may be found.

Table 1 shows a summary of the results for all puz-

zles in the test set, arranged by solution method. This

table shows a consistent drop in average iterations and

average time across methods, but examination of the

other results indicates that the cell ordering has the

most signiﬁcant impact on reducing the number of

iterations. The addition of the quasi-magic pruning

rules is delivered at a smaller overhead in processing

time than the fewest-candidate cell ordering, but the

average and median iterations would support the view

that both modiﬁcations are worth implementing. The

changes in the average time per iteration (measured in

milliseconds) are clearly slight. For some puzzles re-

quiring larger numbers of iterations to enumerate the

search space, the use of fewest-candidate cell order-

ing actually increases the number of iterations, but the

number is reduced for the majority of puzzles. This

behaviour is not observed when the quasi-magic prun-

ing rules are also added.

Through the additional analysis of the breakdown

of results into categories of numbers of givens (Ta-

ble 2), it is clear that the combination of quasi-magic

pruning rules and fewest-candidate cell ordering is by

far the best method. The number of puzzles in each

category of number of givens is shown in parenthe-

sis. Irrespective of whether there is an entirely uni-

form correlation between the number of givens and

the ease with which a puzzle may be solved by hand

(Jones et al., 2007), it seems clear that a strong cor-

relation exists with the speed with which this solver

enumerates the non-pruned search spaces of Quasi-

Magic Sudoku puzzles. The processing overhead of

the quasi-magic pruning rules starts to become appar-

ent on the fastest enumerating puzzles, but not enor-

mously so; the addition of fewest-candidate cell or-

dering seems to overcome this.

Table 3 shows more detailed results, categorised

by the number of givens, for the most successful

method - quasi-magic pruning rules with fewest-

candidate cell ordering. The worst case in the test

set, a puzzle having 6 givens, took 51,594 iterations

(which corresponded to 1.2986 seconds) to fully enu-

merate the non-pruned search space. The overwhelm-

ing majority of the puzzles in the test set required less

than 0.1 seconds. The average time per iteration is re-

markably constant for all but those puzzles requiring

very few iterations. For the declared application, it is

sufﬁcient to know that the non-pruned search spaces

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252

Table 2: Average time (s) for all solution methods, by number of givens.

No q-m No q-m With q-m With q-m

pruning rules pruning rules pruning rules pruning rules

row-by-row fewest-candidate row-by-row fewest-candidate

<7 (12) 7.8336 3.7024 2.5663 0.4451

7-8 (49) 3.3064 1.8388 1.1547 0.0593

9-10 (35) 1.1229 0.1879 0.4348 0.0205

11-12 (45) 0.1706 0.0485 0.0710 0.0057

13-14 (29) 0.0511 0.0130 0.0274 0.0030

>14 (10) 0.0159 0.0043 0.0111 0.0022

Table 3: Iterations and average time per iteration of the method employing quasi-magic pruning rules and fewest-candidate

cell ordering, by number of givens.

number of average maximum median average time per

Givens iterations iterations iterations iteration (ms)

<7 (12) 17,984 51,594 13,051 0.0248

7-8 (49) 2,351 21,586 1,119 0.0252

9-10 (35) 796 7,006 300 0.0258

11-12 (45) 186 933 140 0.0309

13-14 (29) 80 303 52 0.0377

>14 (10) 41 108 30 0.0529

of all puzzles can rapidly be fully enumerated.

The algorithm described in Section 4 is very efﬁ-

cient. Due to the compact representation of the grid

and the early rejection of unpromising states, the im-

plementation of the algorithm has very low memory

requirements; it requires less than 100KB of work-

ing RAM to run. (The updating of the ﬂag vectors of

cells, through bitmasking, is greatly faster than, for

example, the indexing of candidate value information

stored as elements in an array.) This approach is also

scalable, enabling the prospect of similar acceptable

results being achievable for larger Quasi-Magic Su-

doku puzzles, for example of size 64.

6 CONCLUSIONS

This paper demonstrates that domain information for

Quasi-Magic Sudoku puzzles, for the case ∆ = 2, may

be used to reliably and rapidly reconstruct a grid,

from a small set of given values. The knowledge-rich

quasi-magic pruning rules and the proposed cell or-

dering both contribute to a massive reduction in the

size of the search space that needs to be enumerated,

both to determine a solution and, importantly here, to

be certain of the uniqueness of that solution.

In a general sense, the above ﬁndings establish

the applicability of Quasi-Magic Sudoku for the con-

struction of erasure correction codes. This is an im-

portant result, given that Sudoku codes have already

been established as being of interest (Soedarmandji

and McEliece, 2007), and Quasi-Magic Sudoku car-

ries the prospect of grid reconstruction (and hence

message reconstruction) from a smaller set of initial

given values. However, to prove the usefulness of

Quasi-Magic Sudoku in erasure correction, it is nec-

essary also to consider reconstruction from a set of

values which may not correspond well to the minimal

set of independent givens of a particular grid.

As additional further work, we propose the exami-

nation of other cases of ∆, to determine corresponding

domain-rich cell value interrelationships and associ-

ated pruning rules. Other cell orderings, derived from

an understanding of the puzzle, might yield further re-

ductions in the search space. The determination of the

numbers of valid grids for other cases of ∆ would also

be of mathematical interest. The examination of puz-

zles of larger sizes might also be useful within cod-

ing theory and other applications. Lastly, it is pro-

posed that an analysis is performed of how frequently

each of the quasi-magic pruning rules of Section 3 is

employed in pruning the search space, and which are

most effective for the least cost in processing time.

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