Mathematics of Sudoku

8/2/2019 Mathematics of Sudoku

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The Mathematics of Sudoku

Helmer AslaksenDepartment of MathematicsNational University of Singapore

[email protected]

www.math.nus.edu.sg/aslaksen/
mailto:[email protected]://www.math.nus.edu.sg/aslaksen/http://www.math.nus.edu.sg/aslaksen/mailto:[email protected]


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Sudoku grid

9 rows, 9 columns, 9 3x3 boxes and 81 cells

I will refer to rows, columns or boxes as units

(p,q) refers to row p and column q I number the boxes left to right, top to bottom


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Rules

Fill in the digits 1 through 9 so that everynumber appears exactly once in every unit(row, column and box)

Some numbers are given at the start toensure that there is a unique solution


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History of Sudoku

Retired architect Howard Garns ofIndianapolis invented a game calledNumber Place in May 1979

Introduced in Japan in April1984 under the

name of Sudoku (), meaning singlenumbers

Took the UK by storm in late 2004


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Latin squares

In 1783, Euler introduced Latin squares,i.e., n x n arrays where 1 through nappears once in every row and column

A Sudoku grid is a 9x9 Latin square wherethe 9 3x3 boxes contains 1 through 9 once


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How many givens do we need toguarantee a unique solution?

This is an unknown mathematical problem

There are examples of uniquely solvablegrids with 17 givens(www.csse.uwa.edu.au/~gordon/sudokumin.php)


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How many givens can we have

without guaranteeing a unique

solution?2 8 3 6 7 1 9 4 5

9 7 6 5 4 3 1

4 1 5 3 9 7 65 6 7 4 1 9 3 8 2

8 3 4 2 6 7 1 5 9

1 9 2 8 3 5 4 6 73 2 1 7 8 6 5 9 4

7 5 8 9 2 4 6 1 3

6

4

9

1

5

3

7

2

8


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How many Sudoku grids are there?

It was shown in 2005 by BertramFelgenhauer and Frazer Jarvis to be6,670,903,752,021,072,936,960

This is roughly 0.00012% the number of99 Latin squares


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Why Sudoku is simpler than reallife

If a number can only be in a certain cell,then it must be in that cell


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Elementary solution techniques

We will first describe three easytechniques

Scanning (or slicing and dicing)

Cross-hatching

Filling gaps


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Scanning

We can place 2 in (3,2)

You should start scanning in rows or

columns with many filled cells Scan for numbers that occur many times

4 2 8 3

8 1 4 2

7 6 8 5 4


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Cross-hatching


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Filling gaps

Look out for boxes, rows or columns withonly one or two blanks


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Intermediate techniques

The elementary techniques will solve easypuzzles

I will discuss one intermediate technique,box claims a row (column) for a number


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Box claims a row (column) for anumber

Box 1 claims row 1 for number 1


4 2 8 3

8 1 4 2

7 2 6 8 5 4


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Box claims a row (column) for anumber

Box 2 claims row 3 for number 8


This is sometimes called pointingpairs/triples

8 6

5 6 1

4

8

8


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Advanced techniques

For harder puzzles, we must pencil in candidatelists

This is called markup


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Candidate Lists


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Strategy

If you believe the puzzle is easy, youshould be able to solve it using easytechniques and it is a waste of time to

write down candidate lists

If you believe the puzzle is hard, youshould not waste your time with too much

scanning, and go for candidate lists aftersome quick scanning


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Single-candidate cell

5 is the only candidate in (3,3) Called a naked single

169 4589

2 74589

459

35


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Single-cell candidate

(1,2) is the only square in which 6 is acandidate

Called a hidden single

169 4589

2 74589

459

35


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Strategy

Once you fill one cell, you must update allthe affected candidate lists

Search systematically for naked or hiddensingles in all units


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Naked pairs

Cells 2 and 5 only contain 1 and 7

Hence 1 and 7 cannot be anywhere else! We can remove 1 and 7 from the lists in all

the other cells


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Hidden pair

6 and 9 only appear in cells 1 and 5

Hence we can remove all other numbersfrom those two cells, {6, 9} becomes anaked pair and we get a hidden {1}

69 35 357 348 692

578 4781

69 35 357 348 692

578 478 1357

14569

35 357 348 1569 2

578 478 1357


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Naked triples

Cells 2, 3 and 7 only contain a subset of{3, 5, 6}

Hence 3, 5 and 6 cannot be anywhereelse

We can remove 3, 5 and 6 from the lists inall the other cells


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Naked triples

Notice that none of the three cells need tocontain all three numbers

{3, 5, 6} still forms a triple in cells 2, 3 and7 even though all the three lists onlycontain pairs

13458

35 36 3458

1672

56 46789

14679


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Naked and hidden n-tuples

We can generalize the pairs and triples tonaked and hidden n-tuples

If n cells can only contain the numbers

{a1,, an}, then those numbers can beremoved from all other cells in the unit

If the n numbers {a1,, an} are only

contained in n cells in an unit, then allother numbers can be removed fromthose cells


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Naked or hidden?

Naked means that n cells only contain nnumbers

Hidden means that n numbers are onlycontained in n cells

Naked removes the n numbers from othercells

Hidden removes other numbers from the ncells

Hidden becomes naked


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Row (column) claims box for anumber

In the middle row, 2 can only occur in thelast box

Hence we can remove it from all the othercells in the box

Also called box line reduction strategy


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Row (column) claims box for anumber vs. box claims row

(column) for a number Row claims box for a number means that if

all possible occurrences of x in row y are

in box z, then all possible occurrences of xin box z are in row y

Box claims row for a number means that if

all possible occurrences of x in box z arein row y, then all possible occurrences of xin row y are in box z


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More advanced techniques

X-Wing

Swordfish

XY-wing


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X-Wing

We can remove the 6's marked in thesmall squares and we can place 9 in (7,9).


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X-Wing Theory

Suppose we know that x only occurs as acandidate twice in two rows (columns),and that those two occurrences are in the

same columns (rows) Then x cannot occur anywhere else in

those two columns (rows)


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Swordfish

This is just a triple X-wing

Suppose we know that x occurs as acandidate at most three times in threerows (columns), and that thoseoccurrences are in the same columns(rows)

Then x cannot occur anywhere else inthose three columns (rows)


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Swordfish 2

We can place a 2 in (5,2)


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Swordfish 3

We dont need nine candidate lists


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XY-wing

We can eliminate z from the cell with a ?

If there is an x in the top left cell, there has

to be a z in the top right cell If there is a y in the top left cell, there has

to be a z in the bottom left cell


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XY-wing

We dont need a square; it is enough that

there are three cells of the form xy, xz andyz, where the xy is in the same unit as xz

and the same unit yz

We can eliminate z from the gray cellsbelow


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What if youre still stuck?

Sometimes even these techniques dont

work

You may have to apply proof by

contradiction

Choose one candidate in a list, and seewhere that takes you

If that allows you to solve the grid, youhave found a solution


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Proof by contradiction

If your assumption leads to acontradiction, you can strike that numberoff the candidate list in the cell

Unfortunately, you may have to branch at

several cells


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Solution by logic?

Some people do not approve of proof bycontradiction, claiming that it is not logic

It is obviously valid logic, but it is hard todo with pen and paper


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Where can I get help?

There are many Sudoku solvers availableonline

Many of them allow you to step throughthe solution, indicating which techniquesthey are using

http://www.scanraid.com/sudoku.htm
http://www.scanraid.com/sudoku.htmhttp://www.scanraid.com/sudoku.htm


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Warning!

Sudoku is fun, but it is highly addictive

Happy Sudoku!


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Sample Puzzle


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Sample Puzzle 2

Documents

Mathematics of Sudoku