1 / 45

The Mathematics of Sudoku

The Mathematics of Sudoku. Helmer Aslaksen Department 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

brooke
Download Presentation

The Mathematics of Sudoku

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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 units • (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 unit (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?

  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

  9. Why Sudoku is simpler than real life • If a number can only be in a certain cell,then it must be in that cell

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

  11. 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

  12. Cross-hatching

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

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

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

  16. 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”

  17. Advanced techniques • For harder puzzles, we must pencil in candidate lists • This is called markup

  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 units

  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, {6, 9} becomes a naked pair and we get a hidden {1}

  25. 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

  26. 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

  27. 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

  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 • Hidden becomes naked

  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 • Also called “box line reduction strategy”

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

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

  33. 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)

  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 can place a 2 in (5,2)

  36. Swordfish 3 • We don’t need nine candidate lists

  37. 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

  38. 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

  39. 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

  40. 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

  41. 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

  42. 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

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

  44. Sample Puzzle

  45. Sample Puzzle 2

More Related