The Mathematics of Sudoku

1. The Mathematics of Sudoku Helmer AslaksenDepartment of MathematicsNational University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/

2. Sudoku grid • 9 rows, 9 columns, 9 3x3 boxes and 81 cells • I will refer to rows, columns or boxes as areas • (p,q) refers to row p and column q • I number the boxes left to right, top to bottom

3. Rules • Fill in the digits 1 through 9 so that every number appears exactly once in every area (row, column and box) • Some numbers are given at the start to ensure that there is a unique solution

4. 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

5. 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

6. 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)

7. How many givens can we have without guaranteeing a unique solution? • It is possible to have a puzzle with only four cells blank that is still not uniquely determined

8. The Number of Sudoku grids • The number of ways of filling in a blank Sudoku grid was shown in May 2005 by Felgenhauer and Jarvis to be 6,670,903,752,021,072,936,960  

9. Elementary solution techniques • We will first describe three easy techniques • Scanning (or slicing and dicing) • Cross-hatching • Filling gaps

10. 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 time

11. Cross-hatching

12. Filling gaps • Look out for boxes, rows or columns with only one or two blanks

13. Intermediate techniques • The elementary techniques will solve easy puzzles • I will discuss one intermediate technique, box claims a row/column for a number

14. Box claims a row/column for a number • Box 1 claims row 1 for 1 • We can place 1 in (3,8)

15. Box claims a row/column for a number • Box 2 claims 8 in row 3 • We can place 8 in (2,9)

16. Box claims a row/column for a number • If x can only occur in row/column y in box z, then x can only occur in box z in row/column y

17. Advanced techniques • For harder puzzles, we must pencil in candidate lists

18. Candidate Lists

19. 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

20. Single-candidate cell • 5 is the only candidate in (3,3) • Called a naked single

21. Single-cell candidate • (1,2) is the only square in which 6 is a candidate • Called a hidden single

22. Strategy • Once you fill one cell, you must update all the affected candidate lists • Search systematically for naked or hidden singles in all areas

23. 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

24. Hidden pair • 6 and 9 only appear in cells 1 and 5 • Hence we can remove all other numbers from those two cells, and we get a naked {6, 9} pair and a hidden {1}

25. Naked triplets • 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

26. Naked triplets • Notice that none of the three cells need to contain all three numbers • {3, 5, 6} still forms a triplet in cells 2, 3 and 7 even though all the three lists only contain pairs

27. Naked and hidden n-tuples • We can generalize the pairs and triplets to naked and hidden n-tuples • If n candidate lists in an area only contain {a1,…, an}, then those numbers can be removed from all other lists in the area • If {a1,…, an} are only contained in n candidate lists in an area, then all other numbers can be removed from those lists

28. 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

29. 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

30. 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

31. Row/column claims box for a number vs. box claims row/column for a number • All possible occurrences of x in row/column y are in box z if and only if all possible occurrences of x in box z are in row/column y

32. More advanced techniques • X-Wing • Swordfish • XY-wing

33. X-Wing • 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)

34. 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)

35. Swordfish 2 • We don’t need nine candidate lists

36. 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

37. 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 area as xz and the same area yz • We can eliminate z from the gray cells below

38. 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

39. 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

40. 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

41. 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

42. Warning! • Sudoku is fun, but it is highly addictive • Happy Sudoku!

43. Sample Puzzle

44. Sample Puzzle 2