The Mathematics of Sudoku

Helmer AslaksenDepartment of Mathematics

National University of Singapore

aslaksen@math.nus.edu.sg

www.math.nus.edu.sg/aslaksen/

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

Rules

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

• Some numbers are given at the start to ensure that there is a unique solution

History of Sudoku

• Retired architect Howard Garns of Indianapolis invented a game called “Number Place” in May 1979

• Introduced in Japan in April1984 under the name of Sudoku (数独 ), meaning single numbers

• Took the UK by storm in late 2004

Latin squares

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

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

How many givens do we need to guarantee a unique solution?

• This is an unknown mathematical problem

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

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 6

5 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 7

3 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

How many Sudoku grids are there?

• It was shown in 2005 by Bertram Felgenhauer and Frazer Jarvis to be 6,670,903,752,021,072,936,960

• This is roughly 0.00012% the number of 9×9 Latin squares

Why Sudoku is simpler than real life

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

Elementary solution techniques

• We will first describe three easy techniques

• Scanning (or slicing and dicing)

• Cross-hatching

• Filling gaps

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

Cross-hatching

Filling gaps

• Look out for boxes, rows or columns with only one or two blanks

Intermediate techniques

• The elementary techniques will solve easy puzzles

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

Box claims a row (column) for a number

• Box 1 claims row 1 for number 1

• We can place 1 in (3,8)

4         2 8 3  

  8   1   4     2

7 2 6   8   5   4

Box claims a row (column) for a number

• Box 2 claims row 3 for number 8

• We can place 8 in (2,9)

• This is sometimes called “pointing pairs/triples”

8         6

5 6 1

4

8

8

Advanced techniques

• For harder puzzles, we must pencil in candidate lists

• This is called markup

Candidate Lists

Strategy

• If you believe the puzzle is easy, you should be able to solve it using easy techniques and it is a waste of time to write down candidate lists

• If you believe the puzzle is hard, you should not waste your time with too much scanning, and go for candidate lists after some quick scanning

Single-candidate cell

• 5 is the only candidate in (3,3)

• Called a naked single

169 4589

2 74589

459

35

Single-cell candidate

• (1,2) is the only square in which 6 is a candidate

• Called a hidden single

169 4589

2 74589

459

35

Strategy

• Once you fill one cell, you must update all the affected candidate lists

• Search systematically for naked or hidden singles in all units

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

Hidden pair

• 6 and 9 only appear in cells 1 and 5

• Hence we can remove all other numbers from those two cells, {6, 9} becomes a naked pair and we get a hidden {1}

69 35  357  348 69  2 578 478 1

69 35  357  348 69  2 578 478 1357

14569

35  357  348 1569  2 578 478 135

7

Naked triples

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

• Hence 3, 5 and 6 cannot be anywhere else

• We can remove 3, 5 and 6 from the lists in all the other cells

Naked triples

• Notice that none of the three cells need to contain all three numbers

• {3, 5, 6} still forms a triple in cells 2, 3 and 7 even though all the three lists only contain pairs

13458

35  36  3458

167  2 56 46789

 14679

Naked and hidden n-tuples

• We can generalize the pairs and triples to naked and hidden n-tuples

• If n cells can only contain the numbers {a1,…, an}, then those numbers can be removed from all other cells in the unit

• If the n numbers {a1,…, an} are only contained in n cells in an unit, then all other numbers can be removed from those cells

Naked or hidden?

• Naked means that n cells only contain n numbers

• Hidden means that n numbers are only contained in n cells

• Naked removes the n numbers from other cells

• Hidden removes other numbers from the n cells

• Hidden becomes naked

Row (column) claims box for a number

• In the middle row, 2 can only occur in the last box

• Hence we can remove it from all the other cells in the box

• Also called “box line reduction strategy”

Row (column) claims box for a number 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 x in box z are in row y

• Box claims row for a number means that if all possible occurrences of x in box z are in row y, then all possible occurrences of x in row y are in box z

More advanced techniques

• X-Wing

• Swordfish

• XY-wing

X-Wing

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

X-Wing Theory

• Suppose we know that x only occurs as a candidate 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)

Swordfish

• This is just a triple X-wing

• Suppose we know that x occurs as a candidate at most three times in three rows (columns), and that those occurrences are in the same columns (rows)

• Then x cannot occur anywhere else in those three columns (rows)

Swordfish 2

• We can place a 2 in (5,2)

Swordfish 3

• We don’t need nine candidate lists

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

XY-wing

• We don’t need a square; it is enough that there are three cells of the form xy, xz and yz, where the xy is in the same unit as xz and the same unit yz

• We can eliminate z from the gray cells below

What if you’re still stuck?

• Sometimes even these techniques don’t work

• You may have to apply “proof by contradiction”

• Choose one candidate in a list, and see where that takes you

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

Proof by contradiction

• If your assumption leads to a contradiction, you can strike that number off the candidate list in the cell

• Unfortunately, you may have to “branch” at several cells

Solution by “logic”?

• Some people do not approve of proof by contradiction, claiming that it is not logic

• It is obviously valid logic, but it is hard to do with pen and paper

Where can I get help?

• There are many Sudoku solvers available online

• Many of them allow you to step through the solution, indicating which techniques they are using

• http://www.scanraid.com/sudoku.htm

Warning!

• Sudoku is fun, but it is highly addictive

• Happy Sudoku!

Sample Puzzle

Sample Puzzle 2