Farther beyond sudoku Using new puzzles to develop students’ logical-thinking skills Jeffrey Wanko Miami University NCTM Annual meeting - April 10, 2008 NCTM 2007 “Beyond Sudoku” Introduced: Bridges (Hashiwokakero) Fences (Slitherlink) Dominoes Battleships an Original title? September 2007 MTMS - October 2006 October 2006 nctm 2008 “Farther Beyond Sudoku” Introducing: Masyu Shikaku Nurikabe Hitori nikoli Japanese publisher specializing in logic puzzles (language/ culture independent puzzles) Developed and published a number of new puzzle types Popularized Sudoku puzzles (although they first appeared in an American puzzle magazine in 1979 as “Number Place” puzzles) Nikoli.com features free puzzles for everyone to access and contains a “members” area (cost: 550 yen/month) where 4 new puzzles are published each day Today’s four puzzle types were invented by Nikoli MASYU Masyu = “evil influence” Original title (“pearl necklace” in 1998) and subsequent title (“white pearls and black pearls” in 2000) were lengthy. A misreading lead to the name Masyu which has stuck. A rare puzzle type that contains no letters or numbers MASYU Goal: Complete a closed loop that passes through all of the squares with circles (the white and black “pearls”) and that goes through the center of squares horizontally and/or vertically • As the path travels through white circles, it must go straight through, but it must turn in the previous and/or the next square in the path • As the path travels through black circles, it must make a right-angle turn and go straight through the previous square and the next square in the path MASYU • With a Masyu puzzle, where can you start? • Using logic, what do you know has to be true? • Are there some general rules that you can formulate that will work on this and other Masyu puzzles? Masyu Example 1 MASYU • Are there general rules for solving Masyu that this solution has helped you think about? • Do Masyu puzzles support any other mathematical concepts? Masyu Example 1 - Solution MASYU Masyu Example 2 Masyu Example 3 MASYU Masyu Example 2 - Solution Masyu Example 3 - Solution MASYU Masyu Example 4 MASYU Masyu Example 4 - Solution MASYU Masyu Example 5 - 36 x 20 NCTM TRIPLE CROWN RACE #1 - THE UTAH DERBY THE UTAH DERBY Each jockey picks a gate number from 1-100 (inclusively) for your horse to enter The smallest gate number chosen by only one person is the winner Write down your gate on a piece of paper After all gates have been recorded, you must show your gate number to someone else for verification THE UTAH DERBY 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 SHIkaku Shikaku = “four corners” Has appeared in GAMES magazine under the name “Partitions” and in other sources as “Divide by Box” Like crossword puzzles, most Shikaku grids are constructed so that the grid has 180º rotational symmetry (with respect to the numbered and open squares) Shikaku Goal: Subdivide the grid into rectangles (and squares) so that the number in each rectangle refers to the area of that rectangle • Only one number can appear in each rectangle • Each square on the grid is used in exactly one rectangle (no rectangles may overlap) shikaku • With a Shikaku puzzle, where can you start? • Using logic, what do you know has to be true? • Are there some general rules that you can formulate that will work on this and other Shikaku puzzles? Shikaku Example 1 shikaku • Are there general rules for solving Shikaku that this solution has helped you think about? • Do Shikaku puzzles support any other mathematical concepts? Shikaku Example 1 - Solution shikaku Shikaku Example 2 Shikaku Example 3 shikaku Shikaku Example 2 - Solution Shikaku Example 3 - Solution shikaku Shikaku Example 4 shikaku Shikaku Example 4 - Solution shikaku Shikaku Example 5 - 36 x 20 NCTM TRIPLE CROWN RACE #2 - THE PENROSE STAKES THE PENROSE STAKES Each jockey picks a gate number from 1-100 (inclusively) for your horse to enter The smallest gate number chosen by only one person is the winner Write down your gate on a piece of paper After all gates have been recorded, you must show your gate number to someone else for verification THE PENROSE STAKES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 nurikabe Nurikabe: “painting the wall”; (from Japanese folklore) an invisible wall that blocks roads, thus delaying travel First appeared in a March 1991 Nikoli puzzle magazine and has since become a mainstay of Nikoli Appears in some sources as “Islands in the Stream” or “Cell Structure” nurikabe Goal: Create white regions surrounded by black walls. Each white region contains only one number and that number corresponds to the area of that region. The white regions must be separated from each other (but can touch at the corners). • Numbered squares cannot be filled in • The black squares must be linked in a continuous wall • Black squares cannot form a square 2 x 2 or larger nurikabe • With a Nurikabe puzzle, where can you start? • Using logic, what do you know has to be true? • Are there some general rules that you can formulate that will work on this and other Nurikabe puzzles? Nurikabe Example 1 nurikabe • Are there general rules for solving Nurikabe that this solution has helped you think about? • Do Nurikabe puzzles support any other mathematical concepts? Nurikabe Example 1 - Solution nurikabe Nurikabe Example 2 Nurikabe Example 3 nurikabe Nurikabe Example 2 - Solution Nurikabe Example 3 - Solution nurikabe Nurikabe Example 4 nurikabe Nurikabe Example 4 - Solution nurikabe Nurikabe Example 5 - 36 x 20 NCTM TRIPLE CROWN RACE #3 - THE BELL CURVE STAKES THE BELL CURVE STAKES Each jockey picks a gate number from 1-100 (inclusively) for your horse to enter The smallest gate number chosen by only one person is the winner Write down your gate on a piece of paper After all gates have been recorded, you must show your gate number to someone else for verification THE BELL CURVE STAKES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 hitori Hitori = “alone” or “one person” First appeared in a March 1990 Nikoli puzzle magazine Hitori puzzles seem to appeal to those who enjoy Sudoku puzzles but want an additional challenge hitori Goal: Eliminate (fill in) squares so that no number appears more than once in any row or column • Filled in squares cannot be horizontally or vertically adjacent, although they can be diagonally adjacent • The remaining unfilled squares cannot be isolated from the rest of the grid by filled-in squares (that is, you need to be able to move from any unfilled square to another unfilled square) hitori • With a Hitori puzzle, where can you start? • Using logic, what do you know has to be true? • Are there some general rules that you can formulate that will work on this and other Hitori puzzles? Hitori Example 1 hitori • Are there general rules for solving Hitori that this solution has helped you think about? • Do Hitori puzzles support any other mathematical concepts? Hitori Example 1 - Solution hitori Hitori Example 2 Hitori Example 3 hitori Shikaku Example 2 - Solution Shikaku Example 3 - Solution hitori Hitori Example 4 hitori Hitori Example 4 - Solution hitori Hitori Example 5 - 17 x 17 resources Nikoli (online puzzles at www.nikoli.com; examples of puzzles, Japanese magazines and books at www.nikoli.co.jp/en) GAMES and GAMES World of Puzzles magazines (May 2008, GAMES World of Puzzles feature: “New Japanese Logic Puzzles”) Other websites: e.g., www.2n1.org/applets/pearls www.puzzle-nurikabe.com www.hitoriconquest.com what’s next Currently conducting a pilot research study to examine the effects of logic puzzle exposure and practice on students’ logic and spatial reasoning. Serious inquiries about participating in this study next year are welcome! Email me at [email protected] Manuscript based on pilot study underway for “Take Time for Action” department in Mathematics Teaching in the Middle School Manuscript submitted for Mathematics Teacher 2009 Focus Issue on “Proof: Laying the Foundation” Book prospectus being developed—resource book for teachers to use logic puzzles in their classrooms today’s presentation An electronic copy of the handout and a copy of this presentation can be downloaded from my webpage at: www.users.muohio.edu/wankojj.