1. Teaching Students to Invent Puzzles Scott Kim shufflebrain.com Math Fair Workshop April 21, 2007

2. SNAP ¥ Fun math ¥ Present puzzles ¥ Puzzles are assigned

3. Next step: inventing puzzles ¥ Like creating art ¥ Like writing stories ¥ Like math research

4. But how do you invent them? ¥ Where do you get started? ¥ How do you get new ideas? ¥ What makes a good puzzle?

5. I design puzzles

6. Outline What makes a good puzzle? Inventing new puzzles Adapting old puzzles

7. W HAT M AKES A G OOD P UZZLE ? A bit of theory from game design

8. 1. Definition of “Puzzle” A puzzle is fun and has a right answer.

9. 1. Definition of “Puzzle” A puzzle is fun and has a right answer. As opposed to everyday “problems”

10. 1. Definition of “Puzzle” A puzzle is fun and has a right answer. As opposed to everyday “problems” As opposed to a game (no answer) or a toy (no goal)

11. 2. Medium ¥ Spoken ¥ Paper and pencil ¥ Manipulative ¥ Computer

12. 3. Goal ¥ Put together ¥ Take apart ¥ Fill in the blanks ¥ Matching ¥ Get from here to there ¥ Unscramble order ¥ Satisfy conditions

13. 4. Parts

18. 5. Rubric ¥ Attractive (familiar, intriguing) ¥ Simple rules (harmonious, few pieces) ¥ Fun to play (manipulative, unfamiliar) ¥ Good feedback (sense of progress) ¥ Clear goal (pleasing, checkable) ¥ Solvable (deducible, maybe unique)

19. A DAPTING O LD P UZZLES You too can invent puzzles

20. 5 levels of originality 1. Present 2. Adapt 3. Compose 4. Vary 5. Invent

21. 1. PRESENT …an old puzzle

22. Presenting a puzzle ¥ Play lots of puzzles ¥ Choose a puzzle ¥ Present it to someone else ¥ Offer hints as needed

23. 2. REVAMP …an old puzzle in a new way

24. Revamp appearance

25. Revamp theme

26. Revamp context

27. 3. COMPOSE …within an existing form

28. Sudoku ¥ Fill the grid so every row, column and outlined region contains the numbers 1 to 5.

29. Sudoku ¥ Fill in solution ¥ Remove numbers ¥ Solve it ¥ Unique answer?

30. Sudoku — Lessons Learned ¥ Fewer numbers = harder (usually) ¥ May be more than one solution ¥ May be no solution at all ¥ Better if the puzzle has a theme

31. Make a sequence: easy to hard

32. Groups of Levels

33. Levels ¥ Go from easy to hard ¥ Common in computer games ¥ Help player learn the game ¥ Levels work in physical games too

34. Tangrams

35. Logic puzzles

36. Rush Hour

44. 4. VARY …an existing game

45. Start with an existing game

46. Vary difficulty

47. Vary scale

49. Vary size

50. Vary the rules

51. Vary the medium

52. I NVENTING N EW P UZZLES Creative mathematics

53. Getting started ¥ Art — doodle ¥ Writing — what have I experienced? ¥ Machines — what is needed? ¥ Music — what do I care about?

54. Asking the right question 1. What can this do? 2. What’s wrong? 3. What’s the question? 4. How can I generalize? 5. What happens if? 6. How can I make this fun?

55. 1. What can this do?

56. Hint: It’s not the letter L.

57. 1. What can this do? Answer: The letter F.

58. 1. What can this do?

59. 2. What’s wrong? …with Sudoku?

60. 2. What’s wrong? Too abstract…make it physical

61. 2. What’s wrong? Too much time…make it smaller Too repetitive…make shaped regions

62. 2. What’s wrong?

63. 3. What’s the question? ¥ The word TEN is made of 9 sticks. ¥ That’s the answer. What’s question?

64. 3. What’s the question? ¥ Remove six matches and leave ten.

65. 4. How can I generalize? Place 8 queens so none attack each other

66. 4. How can I generalize?

67. ¥ What about 9 queens? ¥ What about other chess pieces? ¥ What about other size boards? ¥ What if queen attacks n others?

68. 4. How can I generalize? ¥ Queens are in pairs ¥ Each pair uses up 3 or more rows/columns ¥ 16 rows/columns ¥ 16/3=5.33 pairs ¥ Therefore, maximum queens=10

69. 4. How can I generalize?

70. 5. What happens if? Roll the ball to the end of the maze.

71. 5. What happens if? What if there were 2 balls instead of 1?

72. 5. What happens if?

73. 6. How can I make it fun? Marcy Cook tiles

74. 6. How can I make it fun? ¥ Attractive (tiles) ¥ Simple rules (place all the tiles) ¥ Fun to play (moving tiles) ¥ Good feedback (use every tile) ¥ Clear goal (use all ten digits) ¥ Solvable (unique solution)

75. Example: Mind Games in Discover ¥ Monthly puzzle for science magazine ¥ One page, three puzzles ¥ About a topic in science or math

76. Creative Process 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

77. Topic: Manipulatives

78. 1. Choose a topic Cuisenaire Rods Pattern Blocks Geoboards Topic: Manipulatives

79. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Build the figure with the ten rods

80. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10

81. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10, 9

82. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

83. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties

84. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties

85. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles Numerical Spatial Logical

86. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

87. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

88. S UMMARY

89. Summary What makes a good puzzle? Definition Medium Goal Parts Rubric Adapting old puzzles 1. Present 2. Revamp 3. Compose 4. Vary 5. Invent Inventing new puzzles What can this do? What’s wrong? What’s the question? How can I generalize? What happens if? How can it be fun?

90. Puzzles = nutritious + delicious Nutritious Nutritious + Delicious

91. Puzzles = fun math MATH PUZZLE Approachable Simple rules Physical Easy to check Meaningful MATH PROBLEM Intimidating Complex rules Abstract Hard to check Meaningless

92. Puzzles = pop math ¥ Puzzles : Math :: Pop songs : music ¥ Short and accessible ¥ At worst they’re time-wasters ¥ At best they’re art

93. Puzzles = art form “A good puzzle can give you all the pleasures of being duped that a mystery story can. It has surface innocence, surprise, the revelation of a concealed meaning, and the catharsis of solution.” — Stephen Sondheim

94. Thank You

95. ¥ Exploring Math Through Puzzles (keypress.com) ¥ Brainteasers Page-a-day Calendar (pageaday.com) ¥ Discover Magazine (discovermagazine.com) ¥ Railroad Rush Hour, Sudoku 5x5 (thinkfun.com) ¥ ThinkFun teacher guides (puzzles.com) ¥ Math dance (mathdance.org) ¥ scottkim.com shufflebrain.com

96. Math Fairs ¥ Like Science Fairs, Reading Fairs ¥ Fun ¥ Big ¥ Public

97. Puzzles = writing math ¥ Goal of math education is literacy ¥ Literacy = reading + writing ¥ What’s missing: creative writing ¥ Inventing puzzles = writing math