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Learn to make your own math puzzles
A hands-on workshop by Puzzle Master Scott Kim
Sponsored by the Julia Robinson Math Festival
As part of the National Math Festival
Washington DC, Apr 22, 2017
HAT IS THIS ABOUT?
In this hands-on workshop you will make your own
versions of two popular mathematical puzzles:
Sudoku and Pentominoes, write your puzzles
down, and give them to other people to solve.
My goal is to give you an experience of
mathematics as a creativity activity where you
don't just solve problems that someone else made
up — you ask your own questions and create
something original. If solving a puzzle is like reading
a book, then this is like a writing workshop.
As a puzzle designer I know that coming up with an
original idea can be daunting, so I start simple —
making a 4 by 4 Sudoku puzzle, and a 5-piece
Pentomino puzzle. For those who want to go
deeper, I offer suggestions for customizing your
puzzle or changing the rules.
HAT MORE CAN I DO?
Kids. Play puzzles and games with your friends. Tell
your parents they’re good for you.
Parents. Read math stories and play puzzles and
games with your kids.
Teachers. Have a puzzle corner in your classroom.
Start the day with puzzles. Have kids invent test
questions — the best way to learn is to teach.
Everyone. Play lots of different kinds of puzzles.
Find out which kinds you like.
HAT IS THE VALUE OF PUZZLES?
Puzzles are the literature of mathematics. Puzzles are
where the ideas in math class come alive, and dance
off the page into our imagination. Traditional math
education is full of worksheets that are boring to fill out
(word problems are third-rate puzzles), and train kids
in very narrow ways of thinking. Puzzle, in contrast, are
exciting to solve, memorable, and require resourceful
think-on-your-feet problem solving.
Puzzles and games are as important for a well-rounded
math education as reading books is to language
education. In an ideal world, math students would
spend much time playing puzzles and games, doing
art projects, building things, and reading math stories.
If that sounds strange, imagine an English class
without books. If English class consisted entirely of
spelling, grammar and verb conjugation, then students
would become proficient at the mechanics of
language, but never fall in love with the subject. They
would ask "when are we ever going to use this?" And
they would not be able to read anything but sentences
written in the forms that appear on their worksheets.
It sounds absurd. But that is exactly how math
education works in most classrooms.
Solving puzzles is a great way to hone your problem
solving skills. Making puzzles makes you think harder,
because you no longer have the safety net of knowing
what the solution is, or even if there is a solution. The
challenge is higher, but so are the rewards. Whatever
your level of skill, making puzzles let you do your own
original math right now.
For more puzzle making resources, go to scottkim.com
RESOURCES
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PUZZLE MAKING
SUDOKU
PUZZLES & RECREATIONAL MATH
Will Shortz Presents The Monster Book of Sudoku
for Kids: 150 Fun Puzzles. St. Martin’s Griffin, 2006.
BedtimeMath.org. Brief daily stories and questions for
kids and parents. Web site and app.
The Art of Sudoku, by Thomas Snyder. Grandmaster Puzzles, 2012. Superior artful puzzles.
The Colossal Book of Mathematics: Classic Puzzles,
Paradoxes, and Problems. By Martin Gardner, W. W.
Norton, 2001. The ultimate recreational math book.
Mutant Sudoku, by Thomas Snyder and Wei-Hwa
Huang, Puzzlewright, 2009. Imaginative variations.
ThinkFun.com. Makers of Rush Hour and other superbly
entertaining physical math puzzles for kids and adults.
Taking Sudoku Seriously, the math behind the
puzzle. By Jason Rosenhouse, Oxford U. Press 2012.
Mathfair.com. How to stage a math fair in your school
where every child makes and hosts a puzzle.
Flatlandthemovie.com. The best math fable ever
written. Now a wonderful animated 30-minute movie.
PENTOMINOES
Pentomino Puzzles: 365 Teasers to Keep Your Brain
in Shape. By Eric Harshbarger, Puzzlewright, 2011.
ABOUT SCOTT KIM
Polyominoes: Puzzles, Patterns, Problems, and
Packings. By Sol Golomb, Princeton U Press 1996.
The math of pentominoes and related puzzles.
Scott Kim is a renowned puzzle designer, author, and
math artist. His book Inversions introduced the world to
ambigrams — words that read upside down. His puzzles have appeared in magazines like Scientific American and Discover, and computer games like Tetris and
Bejeweled. He now designs games for abcmouse.com.
Rainbow Premier Pentominoes. By Learning
Resources. Six plastic sets, in rainbow colors.
Quintillions, Gamepuzzles.com. Superb wood set.
SOLUTIONS
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For more puzzle making resources, go to scottkim.com
PLAYSUDOKU
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PUZZLE MAKING
SUDOKU is one of the world’s most popular math puzzles. Although it uses numbers, Sudoku is a game
of pure logic. Sudoku was invented in 1979, and became a worldwide hit when it appeared in the Times
of London in 2004. Its simple rules and deep strategy make it an enduring classic among puzzles.
Standard Sudoku is played on a 9x9 grid. Here are quicker Sudoku puzzles on smaller grids.
BEGINNER: Fill the grid
so every row, column,
and 2x2 box contains all
the numbers 1 to 4.
INTERMEDIATE: Fill the grid so
every row, column, and outlined region contains all the
numbers 1 to 5.
ADVANCED: Fill the grid so every row,
column, outlined region and the
marked diagonal contains all the
numbers 1 to
HINT: What number
goes on the star? It can’t
be 1 or 2, because those
are in the same box.
HINT: The two stars must be
the numbers 2 and 4, because
the region contains 5, 3, 1.
Which star is a 2, which is 4?
HINT: The star must be a 4, because
no other space along the diagonal
can contain a 4. Fill in the rest of the
diagonal.
GOING DEEPER
Be a puzzle connoisseur! You can get more out playing puzzles by reflecting on your experience.
What do you like about Sudoku? Why is it fun?
What other puzzles does Sudoku remind you of?
What do you not like about Sudoku? How could it be better?
What strategies did you use? What insights did you have?
Can you remove any of the starting numbers and still have a unique solution?
How do the irregularly shaped regions and diagonal circles change how you solve the puzzle?
Now teach someone to play Sudoku!
For more puzzle making resources, go to scottkim.com
Name: _________________________ Date: __________________________
MAKESUDOKU
1
LEVEL
Level 1: Beginner. FILL THE GRID. Start here to make your first Sudoku puzzle.
Step 1. MAKE A SOLUTION
PUT A NUMBER 1 to 4 in any square.
FILL THE REST of the grid so every row, column, and 2x2 box
contains all the numbers 1 to 4.
THIS IS your solution!
Step 2. REMOVE NUMBERS
COPY A FEW numbers from your solution into the same places in
this grid. Underline these “given” numbers to remember them.
TRY SOLVING your puzzle. You want your puzzle to have just one
solution. If you reach a point where two different numbers would
both work, choose one, and underline it as a “given” number.
THE UNDERLINED NUMBERS are your puzzle!
Step 3. TEST YOUR PUZZLE
COPY ONLY the underlined numbers from your solution into
the same places in this grid.
GIVE THIS PUZZLE to someone else to try solving. Tell them
the rules. Help them if they get stuck.
CRITIQUE your puzzle. Were your instructions clear? Was it
too hard or too easy? How could it be better? Send a photo
of your creation to [email protected].
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
PLAYPENTOMINOES
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PUZZLE MAKING
PENTOMINOES are the twelve shapes that can be made by putting five squares together. They are one
of the richest and most fruitful geometric puzzles ever invented. The shapes are labeled with the letters
they resemble. Mathematician Sol Golomb named “pentominoes” as a playful generalization of
“dominos”. Mathematicians have invented thousands of questions about Pentominoes, some unsolved.
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BEGINNER: Make
the rectangle using
the three shapes.
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INTERMEDIATE: Make the 5x5
square using the five shapes.
ADVANCED: Make the 6x10 rectangle
using all twelve Pentomino shapes. I’ve
given you a few shapes to get you started.
GOING DEEPER
What do you like about Pentominoes? Why is it fun?
What strategies did you use to solve the puzzles? What insights did you have?
What other puzzles or games use shapes similar to Pentominoes?
How many different shapes can you make with 4 squares? 6?
Now teach someone to play Pentominoes!
For more puzzle making resources, go to scottkim.com
Name: _________________________ Date: __________________________
MAKEPENTOMINOES 1
LEVEL
Level 1: Beginner. CHOOSE THE PIECES. Start here to make your first Pentominoes puzzle.
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Step 1. MAKE A SOLUTION
DIVIDE THE SQUARE into five different shapes, each
containing five squares. This is the solution to your puzzle!
Circle the five shapes you used.
CHECK YOUR PUZZLE to make sure it does not have any
alternate solutions. If it does, choose different pieces.
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Step 2. TEST YOUR PUZZLE
CIRCLE the five pieces used in your solution. Leave the grid
blank. This is your puzzle!
GIVE THIS PUZZLE to someone else to try solving. Tell them
the rules. Help them if they get stuck.
CRITIQUE your puzzle. Were your instructions clear? Was it
too hard or too easy? How could it be better? Send a photo
of your creation to [email protected].
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
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TAKE A
HANDOUT
Puzzles are the
literature of
mathematics.
We read to our kids to create a life-long love of books.
We should do the same for math. Puzzles play the same
role in math education as books do in language education. Play puzzles and games with your kids to create a
life-long love of math and problem solving. Make puzzles with your kids to explore the creative side of math.
“Recreational mathematics should be regularly introduced as a way to interest young students in the wonders of mathematics.” — Martin Gardner, science writer
TAKE A
HANDOUT
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MAKING
ESUDOKU
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PLAY
Play a quick round of the world’s most
popular number puzzle.
Choose from three levels of difficulty:
4x4, 5x5, or 6x6. Or play a puzzle that
someone across the table has made.
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PUZZLE MAKING
Then make a Sudoku puzzle on the
other side of this table.
MAKE
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PUZZLE MAKING
SUDOKU
Make your own original Sudoku puzzle.
Choose from three levels of originality:
fill in a template, vary the board, or
change the rules.
Then test your puzzle by handing it to
someone across the table.
PLAY
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PUZZLE MAKING
PENTOMINOES
Play a quick round of the world’s most
versatile geometric puzzle.
Choose from three levels of difficulty:
5x3, 5x5 or 10x6. Or play a puzzle that
someone across the table has made.
Then make a pentomino puzzle on the
other side of this table.
MAKE
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PUZZLE MAKING
PENTOMINOES
Make your own pentomino puzzle.
Choose from three levels of originality:
choose pieces, draw the outline, or pose
a challenge
Then test your puzzle by handing it to
someone across the table.
PLAYSUDOKU
1
LEVEL
TO DO. Use the number disks to fill the grid so every row, column,
and outlined region contains every number 1 to 4.
HINT. You never have to guess. You can figure out every number
by thinking carefully. For instance, what number can go on the
star? It can’t be a 1 or 2, because those numbers are in the same
outlined region. And it can’t be a 3 because that number is in
the same column.
GOING DEEPER
What do you like about Sudoku? Why is it fun?
What do you not like about Sudoku? How could it be better?
What strategies did you use? What insights did you have?
Now teach someone to play Sudoku, or go make a puzzle.
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
PLAYSUDOKU
2
LEVEL
TO DO. Use the number disks to fill the grid so every row, column,
and outlined region contains every number 1 to 5. Hint: no guessing required — use logic to deduce numbers.
HINT. What numbers go on the two stars?
How do you know?Lorem ipsum dolor sit amet, consec
GOING DEEPER
Which two regions are the same shape in this Sudoku board?
The two circled spaces must contain the same number. Why?
Which starting number could have been removed?
Now teach someone to play Sudoku, or go make a puzzle.
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
PLAYSUDOKU
3
LEVEL
TO DO. Use number disks to fill the grid so every row, column, outlined
box, and the marked diagonal contains every number 1 to 6.
GOING DEEPER
What strategies did you use that involved the diagonal?
Does the diagonal make this Sudoku puzzle easier or harder?
Which number could be removed so the solution is still unique?
Now teach someone to play Sudoku, or go make a puzzle.
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
MAKESUDOKU
1
LEVEL
FILL IN A TEMPLATE. The easy way to make a Sudoku puzzle.
• Fill the blank grid with number disks so every row, column,
and outlined region contains all the numbers 1 to 4. This
is the solution. Write it down on page 3 of the handout.
• Remove as many numbers as you can, making sure that
the puzzle still has only one solution.
• Write your puzzle on page 3 of the handout and give it
to someone to try solving. Improve the puzzle if needed.
Make sure the puzzle: has one and only one solution, and
has clear instructions. How can you improve your puzzle?
GOING DEEPER
• How is making a puzzle different from solving a puzzle?
• How do you know whether the solution is unique?
• How few starting numbers can your puzzle have?
Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
MAKESUDOKU
2
LEVEL
DRAW THE BOXES. An advanced way to make a Sudoku puzzle.
• Use dry erase marker to outline 5 areas of 5 squares each.
• Fill the grid so every row, column, and outlined region
contains all the numbers 1 to 5. Write down your solution.
• Remove numbers, keeping the solution unique.
• Write your puzzle, give it to someone to try. Does it work?
GOING DEEPER
Is it possible for the five regions to all be the same shape?
Is it possible for the five regions to all be different shapes?
Is it possible for just four of the regions to be the same shape?
Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
MAKESUDOKU
3
LEVEL
CHANGE THE RULES. Make your own original Sudoku variation.
• Use dry erase marker to outline 6 areas of 6 squares each.
• Circle six squares, no two of which are in the same row or same
column. Make a rule that these numbers must all be the same, or
that they must all be different. Or make up your own rule.
• Fill in numbers to make a solution. Write it down. If no solution
is possible, alter region shapes or circled squares and try again.
• Remove numbers, keeping the solution unique.
• Write your puzzle, including the new rule, have someone try it.
Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
PLAYPENTOMINOES 1
LEVEL
PENTOMINOES are the shapes that can be made by assembling five
squares. “Pento” = five, “mino” as in “domino”. The shapes are
named after the letters they resemble. Pentominoes were invented by
mathematician Sol Golomb, have many mathematical questions.
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TO DO. Make these shapes by assembling these pieces.
Fill this rectangle
using these pieces:
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Fill this shape using
these pieces:
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GOING DEEPER
Which of these puzzles was harder? Why?
Can you find another way to make a 5x3 rectangle using Pentominoes?
What other games use shapes similar to Pentominoes?
What strategies did you use? How did you decide where to start?
Now teach someone to play pentominoes, or go make a puzzle.
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
PLAYPENTOMINOES 2
LEVEL
PENTOMINOES are the shapes that can be made by assembling five
squares. “Pento” = five, “mino” as in “domino”. The shapes are
named after the letters they resemble. Pentominoes were invented by
mathematician Sol Golomb, have many mathematical questions.
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TO DO. Make a 5 by 5 square by assembling these pieces.
Fill this square
using these pieces:
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Fill this square
using these pieces:
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GOING DEEPER
Which of these puzzles was harder? Why?
What strategies did you use? Which piece was hardest to use?
Can you find another way to make a 5x5 square using Pentominoes?
Can you make a 5x5 square using 3 of one shape, and 2 of another?
Now teach someone to play pentominoes, or go make a puzzle.
For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
PLAYPENTOMINOES 3
LEVEL
TO DO. Make a 10 by 6 rectangle by assembling all twelve Pentomino
pieces. I’ve given you a few pieces to get you started. Then teach
someone to play pentominoes, or go make a puzzle.
EASY
Place Pentominoes on the
four colored areas.
Then fill the rest of the rectangle
using the other eight Pentominoes.
HARD
Place pieces on the
three colored areas.
Then fill the rest of
the rectangle using
the other eight
Pentominoes.
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
MAKEPENTOMINOES 1
LEVEL
YOU DRAW THE OUTLINE.
• Make a shape in the yellow area using 3 or 4 Pentomino
pieces. Use 3 pieces for an easy puzzle, 4 for a hard puzzle.
• Outline your shape using a dry erase marker.
This is your solution! Write it down on a piece of paper.
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Slide the pieces down into the blue area, away from the puzzle.
Give the puzzle to someone to solve. Give them a hint if they
need one. Improve your puzzle until you are happy with it
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Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
MAKEPENTOMINOES 2
LEVEL
YOU CHOOSE THE PIECES.
• Find your own way to make a 5x5 square using five pentomino
pieces. This is your solution. Write it down.
• Check to make the solution to make sure it is unique.
• Write down your puzzle, circling the pieces to use, and give it to
someone to try solving. Improve the puzzle if needed.
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GOING DEEPER
Is it easier to solve a puzzle or make a puzzle? Why?
Pick three random pieces. Can you make a square
with these and two more pieces that you choose?
What other rectangles can you make using Pentominoes?
Now teach someone to play pentominoes, or go make a puzzle.
Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
MAKEPENTOMINOES 3
LEVEL
YOU PLAN THE HINTS.
• Here are six solutions for cutting a 6 by 10 rectangle into all
twelve Pentominoes. Choose one rectangle to be your solution.
• Draw a few of the pieces from the solution in the big blank grid,
already in the right position. Choose pieces so there is only
one way to fit the rest of the pentominoes to fill the rectangle.
• Cover the solutions above and test your puzzle on someone else.
Place Pentominoes on
the colored areas.
Then fill the rest of the rectangle
using the rest of the Pentominoes.
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Send a photo of your creation to [email protected] For more puzzle making resources, go to scottkim.com
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PUZZLE MAKING
MYSUDOKU
Name:
Date:
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PUZZLE MAKING
RULES: Fill the grid so every row, column, and outlined region
contains all the numbers 1 to ____.
For more puzzle making resources, go to scottkim.com
Name:
Date:
MYPENTOMINOES
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PUZZLE MAKING
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Challenge:
For more puzzle making resources, go to scottkim.com
Name: _________________________ Date: __________________________
MAKESUDOKU
2
LEVEL
Level 2: Intermediate. DRAW THE BOXES. Customize your puzzle by drawing your own boxes.
Step 1. DRAW THE BOXES
DRAW LINES to divide the grid into five regions, each
containing five squares. Any shapes will work!
FILL THE GRID so every row, column, and outlined region
contains all the numbers 1 to 5.
THIS IS your solution!
Step 2. REMOVE NUMBERS
COPY A FEW numbers from your solution into the same places in
this grid. Underline these “given” numbers to remember them.
TRY SOLVING your puzzle. You want your puzzle to have just one
solution. If you reach a point where two different numbers would
both work, choose one, and underline it as a “given” number.
THE UNDERLINED NUMBERS are your puzzle!
Step 3. TEST YOUR PUZZLE
COPY the given numbers from your solution into this grid.
GIVE THIS PUZZLE to someone. Tell them the rules: fill the
grid so every row, column and outlined region contains every
number 1-5. Watch them solve the puzzle. Help if necessary.
CRITIQUE your puzzle. Were your instructions clear? Was it
too hard or easy? How could you change it to be better?
Send a photo of your creation to [email protected].
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
Name: _________________________ Date: __________________________
MAKESUDOKU
3
LEVEL
Level 3: Expert. CHANGE THE RULES. Get creative! Make up your own version of Sudoku.
Step 1. CHANGE THE RULES
ADD, CHANGE OR DROP A RULE. For instance you can add
a rule like certain squares need to contain the same number.
Or change the shape of the grid. Or drop the regions rule.
MAKE A SOLUTION by filling the grid so the numbers follow
your rules. If you can’t solve it, adjust the rules and try again.
THIS IS your solution!
Step 2. REMOVE NUMBERS
COPY A FEW numbers from your solution into the same places in
this grid. Underline these “given” numbers to remember them.
TRY SOLVING your puzzle. You want your puzzle to have just one
solution. If you reach a point where two different numbers would
both work, choose one, and underline it as a “given” number.
THE UNDERLINED NUMBERS are your puzzle!
Step 3. TEST YOUR PUZZLE
COPY ONLY the underlined numbers from your solution into
the same places in this grid.
GIVE THIS PUZZLE to someone else to try solving. Tell them
your new rules. Help them if they get stuck.
CRITIQUE your puzzle. Were your instructions clear? Was it
too hard or too easy? How could it be better? Send a photo
of your creation to [email protected].
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
Name: _________________________ Date: __________________________
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MAKEPENTOMINOES
LEVEL
Level 2: Intermediate. DRAW THE OUTLINE. Customize your puzzle by drawing your own shape to fill.
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Step 1. DRAW AN OUTLINE
DRAW AN INTERESTING OUTLINE containing
a multiple of five squares: 20, 25, 30, 35, 40, etc.
DRAW LINES to divide the outline into
5-square pieces that are all different. If you
can’t do it, change the outline and try again.
This is your solution!
CHECK YOUR PUZZLE to make sure it does not
have any alternate solutions. If it does, choose
different pieces or a different outline.
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Step 2. TEST YOUR PUZZLE
REDRAW the outline here, and circle the pieces
used in your solution. This is your puzzle!
GIVE THIS PUZZLE to someone else to solve. Tell
them the rules. Help them if they get stuck.
CRITIQUE your puzzle. Were your instructions
clear? Was it too hard or too easy? How could it
be better? Send a photo of your creation to
[email protected].
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING
Name: _________________________ Date: __________________________
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MAKEPENTOMINOES
LEVEL
Level 3: Expert. CHANGE THE RULES. Get creative! Make up your own Pentominoes challenge.
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Step 1. POSE A CHALLENGE
POSE A CHALLENGE using some or all of the
pentomino pieces. You don’t have to know if it
is solvable! Write down your challenge.
FOR INSTANCE: Make two 6x5 rectangles using
all 12 pentominoes. Make a symmetrical shape
by joining the N and V pieces. Which pentominoes can you fold into a cube minus one side?
SOLVE YOUR CHALLENGE. Adjust the question if it is too easy or too hard.
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Step 2. TEST YOUR PUZZLE
DRAW and write your puzzle here, and circle the
pieces needed. This is your puzzle!
GIVE THIS PUZZLE to someone else to solve. Tell
them the challenge. Help them if they get stuck.
CRITIQUE your puzzle. Were your instructions
clear? Was it too hard or too easy? How could it
be better? Send a photo of your creation to
[email protected].
For more puzzle making resources, go to scottkim.com
!
PUZZLE MAKING