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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
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