1. Mathematics as a Creative Art
Scott Kim
Iolani School • February 4, 2008

2. What’s missing English mechanics = grammar
English creative = writing original work
Math mechanics = algorithms
Math creative = ?

3. Mathematics as a creative art
Creating original, expressive work
Like art or writing
Like math research
The more you invent, the deeper your thinking.

4. But how can students create math?
Original creative work
Engaging, meaningful
Every student succeeds
Danger: designing puzzles that are just fancy textbook problems

5. Answer: Puzzles Puzzles are math made fun Something for every ability
Many opportunities for creativity
Give you some perspective on why puzzles are valuable.

6. Math Fairs (mathfair.com)
Students build puzzles
Add themes of their own
Present in fair-like setting

7. Game Clubs (thinkfun.com)
Pack of 6 puzzles, 4 copies each
Students work at stations
Reflect on strategies
Teacher manuals

8. How do you create puzzles?
Where do you get started?
How do you get new ideas?
What makes a good puzzle?
Danger: designing puzzles that are just fancy textbook problems

9. I design puzzles

10. Railroad Rush Hour Published by ThinkFun
Sequel to Rush Hour I designed
Rush Hour Extravaganza is a Game Club pack

11. 1. Compose challenges Work backwards Add pieces to board
Compose sequence easy to hard

12. 2. Change presentation
Change size
Change appearance
Change story

13. 3. Vary rules
Vary board size
Vary pieces
Vary goal

14. Summary Creative math = puzzles Math Fairs, Game Clubs
1. Compose challenges
2. Change presentation
3. Vary rules

15. Thank You

16. Thank You scottkim.com shufflebrain.com
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

17. What makes a good puzzle? Adapting old puzzles Inventing new puzzles
Outline
What makes a good puzzle?
Adapting old puzzles
Inventing new puzzles
Today’s talk. Behind the scenes. How In invented them.

18. WHAT MAKES A GOOD PUZZLE?
It helps to have some language for talking about puzzle design.
A bit of theory from game design

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

20. 1. Definition of “Puzzle”
A puzzle is fun and has a right answer.
As opposed to everyday “problems”
The key difference. Subjective.

21. 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)
Versus other play.

22. 2. Medium Spoken Paper and pencil Manipulative Computer
Will Shortz radio puzzle, two pencil, body as manipulative, nonphysical computer puzzles.

23. 3. Goal Put together Take apart Fill in the blanks Matching
Get from here to there
Unscramble order
Satisfy conditions
Always something primally pleasing.

24. 4. Parts
Schematically, a puzzle challenges the player to get from a problem to a solution.

25. 4. Parts
But of course the path is never simple. Every puzzle requires that the player make choices, some of which lead to dead ends.

26. 4. Parts
Puzzles in a game have a larger situation that gives the puzzle meaning. Applying the solution lets you move forward in the game.

27. 4. Parts
Good puzzles have require insight. The insight above is to walk around the outside of the maze. Obscure insights, however, feel unfair.

28. 4. Parts
Different puzzles emphasize different parts of the journey. Persistence puzzles are a slow steady climb. Aha! Puzzles skip the climb and go straight to the insight. Story puzzles work the setup into the story. Crossword puzzles are full of little insights: each word unlocks more.

29. 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)
Attractive (familiar, intriguing)
Simple rules (few elements)
Fun to play (novel thought pattern)
Good feedback (sense of progress)
Clear goal (pleasing, checkable)
Solvable (deducible, maybe unique)

30. You too can invent puzzles
ADAPTING OLD PUZZLES
Like any creative act. How do you write a song? Paint a picture?
Some of it is technique. Some of it is art.
To do it well takes work. But to get started is open to everyone.
You too can invent puzzles

31. 5 levels of originality 1. Present 2. Adapt 3. Compose 4. Vary
5. Invent
Natural progression.

32. 1. PRESENT
…an old puzzle
The most basic. Like writing new words to a familiar song.

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

34. …an old puzzle in a new way
2. REVAMP
…an old puzzle in a new way
The most basic. Like writing new words to a familiar song.

35. Revamp appearance

36. Revamp theme

37. Revamp context

38. …within an existing form
3. COMPOSE
…within an existing form
The most basic. Like writing new words to a familiar song.

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

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

41. 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

42. Make a sequence: easy to hard

43. Groups of Levels

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

45. Tangrams

46. Logic puzzles

47. Rush Hour

48. Rush Hour

49. Rush Hour

50. Rush Hour

51. Rush Hour

52. Rush Hour

53. Rush Hour

54. Rush Hour

55. 4. VARY
…an existing game

56. Start with an existing game

57. Vary difficulty

58. Vary scale

59. Vary scale

60. Vary size

61. Vary the rules

62. Vary the medium

63. INVENTING NEW PUZZLES
Creative mathematics

64. Getting started Art — doodle Writing — what have I experienced?
Machines — what is needed?
Music — what do I care about?
Art -- start with technique, or start with inspiration. Clearly making something.
Writing -- write about what you know
Machines -- what do people need? How can I do it better?
Music -- levels of originality. Change the words, to much deeper. Puzzles are linke songs.

65. 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?

66. 1. What can this do?

67. Hint: It’s not the letter L.
1. What can this do?
Hint: It’s not the letter L.

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

69. 1. What can this do?

70. 2. What’s wrong?
…with Sudoku?

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

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

73. 2. What’s wrong?

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

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

76. Place 8 queens so none attack each other
4. How can I generalize?
Place 8 queens so none attack each other
I got interested in puzzles by reading Martin Gardner’s column in Scientific American.

77. 4. How can I generalize?
I got interested in puzzles by reading Martin Gardner’s column in Scientific American.

78. 4. How can I generalize? What about 9 queens?
What about other chess pieces?
What about other size boards?
What if queen attacks n others?

79. 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

80. 4. How can I generalize?

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

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

83. 5. What happens if?

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

85. 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)

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

87. 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

88. Topic: Manipulatives

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

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

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

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

93. 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

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

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

96. 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

97. 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

98. 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

99. SUMMARY

100. What makes a good puzzle? Adapting old puzzles Inventing new puzzles
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?
Today’s talk. Behind the scenes. How In invented them.

101. 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

102. What’s missing ? Goal of math education is literacy
Literacy = grammar + writing
What’s missing: creative math
Mechanics
Creative
English
Grammar
Writing
Math
Computing ?