The Math Games AMTNYS: November 2010

Jessica Kennis

Rebecca Mullen

Jenna Tatu

NCTM:

Use the language of mathematics to express mathematical ideas precisely.

Explore relationships of two-dimensional geometric objects.

Analyze and evaluate the mathematical thinking and strategies of others. MCC:

CM.12 Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

G.RP.1 Recognize that mathematical ideas can be supported by a variety of strategies.

A2.PS.3 Observe and explain patterns to formulate generalizations and conjectures.

a b c d e f g h I

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Foreign Puzzles for the Classroom There are many games from different countries that have mathematical backgrounds and patterns behind them. Many of these can be used in the classroom either to introduce a specific topic, used as extra credit, or simply as material to give when there is a substitute in the classroom for the day. Three specific games are Sudoku and Kakuro.

Sudoku (“Single Number”) Explanation of the game: The objective of this Japanese game is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contain all of the digits from 1 to 9.

Example: Try to fill in this mini Sudoku by using the digits 1-4

Fun Sudoku Facts:

- There number of possible Sudoku grids is slightly less than that of the population of humans on Earth. - You need a few lifetimes to solve all of the Sudoku puzzles. - Unlike crosswords, which need to be adapted for every language, Sudoku can be understood all around the

world. - By playing Sudoku regularly, you can boost your concentration and focus, prevent or ease depression, dementia

and even Alzheimer’s disease according to some studies.

Basic Strategies to Solving Sudoku:

Strategy 1: Use the numbers that already exist to place the missing numbers. -Let’s start by entering a “6” in the shaded square -We will use existing 6’s to eliminate options

-Each square to the right of the shaded square already contains a number 6. -Because each row of nine boxes can only contain one number 6, the second

and third rows of our shaded square cannot be a six. (refer to red line).

-Now we have two possibilities to place the 6. -Looking at the columns: Because columns a and c already contain a 6, this

leaves only one space for us to place our 6. -Therefore we can place our 6 in cell 1b

4 3

3 6 5

5 8 2 6

6 9 8

9 3 6 2

8 7 1

6 9 5 4

4 1 3

1 6

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a b c d e f g h I

Clue

Digit Entry

Strategy 2: Determining the numbers in a column or row. -Row 3 needs a number 3.

-Because right most square already contains a 3, cells 3g, 3h, and 3i cannot have a 3. (Remember: Only one 1-9 per square, column and row).

-Because columns b and c have a 3, this eliminates cells 3b and 3c.

-This leaves just one space left to place our 3. -Therefore we can place our 3 in cell 3e.

There are many more strategies that can be used in solving a Sudoku puzzle. The previous two strategies are just basic strategies. Combinations of these, as well as others can be used in order to solve a Sudoku puzzle. For additional, more complicated strategies, refer to the following websites: http://www.sudokudragon.com/sudokustrategy.htm http://www.sudokuessentials.com/sudoku_tips.html **The math behind this game, used in the classroom it focuses on logic and permutations.

Kakuro (“Cross Sums”) Explanation of the game: Much like a crossword puzzle, the object of the puzzle is to insert a digit from 1-9 into each white cell such that the sum of the numbers in each entry matches the clue associated with it. If the clue is on the upper right triangle, it is associated with the row entries. If the clue is on the lower left triangle, it associates with the column entries.

Example: The clue is 3. The sum of the digits in the entry must equal the clue. Because there are two spaces in the entry, the digits must be

2 and 1. Therefore 2+1=3. Rules: No digit can be duplicated in any entry Each clue must have at least two numbers that add up to it. Example: Try to fill in this simple Kakuro puzzle, using the digits 1-9

4 3

6 5

5 8 2 X X 6

6 9 8

9 3 6 2

8 7 1

6 9 5 4

4 1 3

1 6

Fun Kakuro Facts: - These puzzles resemble crosswords which use numbers instead of words. -Kakuro grids can be any size, though usually the squares within them have to be arranged

symmetrically. -The more blank squares, the harder the puzzle is! Basic Strategies to Solving Kakuro: Strategy 1: Memorize or write down partitions

-Partitioning a number is to break it up into smaller pieces. (Remember, no two of the partitions can be equal in

Kakuro). -A list of Partitions up to entry length of five: (You can have

your classroom come up with the rest of the partitions at a maximum of 9 parts).

Strategy 2: Cross Reference -Find 2 intersecting entries and compare the clue for

combinations of each. Any digits that appear in the combinations for both entries are candidates for the intersection point.

There are many more strategies that can be used in solving

Kakuro puzzles. Refer to the following sites for more information. http://www.conceptispuzzles.com/index.aspx?uri=puzzle/kakuro/techniques http://www.bestkakuro.com/solving.htm

**The solving of Kakuro puzzles can help students with investigating combinations, number partitions and

number order.

Parts Number Partition

2 3 1+2

4 1+3

16 7+9

17 8+9

3 6 1+2+3

7 1+2+4

23 6+8+9

24 7+8+9

4 10 1+2+3+4

11 1+2+3+5

29 5+7+8+9

30 6+7+8+9

5 15 1+2+3+4+5

16 1+2+3+4+6

34 4+6+7+8+9

35 5+6+7+8+9

Race to 100

This is a simple game that can be played in any classroom from second grade on. It’s simple, two players compete to

reach a sum of 100. Player 1 picks any number up to ten. Player 2 can then add up to ten to the number. The game

continues with each player adding up to ten to the number until someone reaches 100. This simple game has a twist!

The first player can always win, if he or she discovers the necessary strategy!

Sample Game John: 3 Mary: 13 John: 21 Mary: 30 John: 38 Mary: 47 John: 57 Mary: 66 John: 75 Mary: 79 John: 82 Mary: 89 John: 97 Mary: 100

To discover the strategy, let’s work backwards. If a player wants to win on his next turn, what number should he pick?

He needs a number where 100 is not obtainable by his opponent, but it is obtainable for his next turn! So, if we call the

necessary number N, we have the following conditions:

However, in the above example, John picked a number greater than 80 and still lost. The player must force his/her

opponent to pick a number between 90 and 99, just like Mary did in her second to last turn! So

is the ideal number for a second to last turn. Any player that chooses 89 is guaranteed to win! So, what about the third

to last turn? Well, if a player wants the opportunity to choose 89 he must force his opponent to say a number between

79 and 88. So, in the third to last turn, choose 78! Do you see the pattern? How could John have ensured his win from

the beginning? First, John must take exactly ten turns! Next, he must vary his answer by 11 each time, so his backwards

game would be:

What can this teach your students? If nothing else, students will see math can be fun! Best case scenario, your students

play enough to see the pattern and use the backwards problem solving technique just as we did. This can also be used

to encourage mental math with non integer values. Try “Race to One”, using fractions or decimals. Each turn a player

can add up to

or .05!

XX X0 0 0

X X 00 X XX 0 0

X X0 0

X 0

Twisted Tic Tac Toe

Most of us had Tic Tac Toe mastered before we left elementary school! Twisted Tic Tac Toe

transforms an old school game into a new challenge with multiple game boards! The object

of the game is still three in a row, but the row cannot be contained on the same game board.

In fact, you could fill up one entire game board with X’s or O’s and still lose! Start with three

game boards. Each board is the side of a triangle; the columns wrap around the entire

triangle and the rows can be created vertically or diagonally across two different boards. As

you master the game, you can simply add another game board! Four boards form a square;

five boards form a box; six boards form a cube!

2-D Cat’s Game to Twisted Triangle:

Two players start by playing on a single game board but have

reached a draw, so they add on the top and bottom boards.

Traditional 3-in-a-rows no longer count and it is no longer

possible to win horizontally.

The players continue their turns as numbered until X traps O.

The highlighted boxes represent the winning wrap-around-row. X has

won the game!

Be Square:

Add a fourth game board that will fold into a square!

The rules remain the same, except, this time you must have 2 rows to win!

It is possible to use a space twice, but only one space may be used to create the two winning rows.

Tic Tac Torus:

Up the challenge with five game boards and complete three rows to win!

This time, horizontal rows can be made across two game boards.

On the torus, it is very easy to create two-in-a-row so that the third can be placed in two separate

spaces. In this case, don’t try to block your opponent; try to create your own row.

Once three-in-a-row has been completed, put a line through that row. Those spaces are now out of

play; none of the three spaces may be used to create another three-in-row.

Below is a sample game won by “0”. The spaces were played in number order and the winning rows

have been color coded.

X0 0 X

0 X X X 0X 0 X

X 0 X 0 X

X 0 00 X

0 0

Class time is a precious commodity. Why play Twisted Tic Tac Toe?

Like most games, it teaches problem solving, but this game helps students see the relationship

between two-dimensional and three-dimensional objects.

By separating the game boards students are forced to envision where each board will meet.

Some interesting facts:

Tic Tac Toe, originally known as Naughts and Crosses, has been played for centuries. Evidence of the

oldest game boards were found all over ancient Rome.

In 1952, as his PhD thesis, Cambridge University student, A.S. Douglas created the world’s very first

computer game; it was a version of Tic Tac Toe.

Martin Gardner found a trick to predict a game of Tic Tac Toe! Watch a video at::

http://wildaboutmath.com/2010/06/25/terrific-tic-tac-toe-trick/

Other versions of Tic Tac Toe that may be useful in your classroom.

Place a math problem in each square. Students will have to solve the problem before they can place

their symbol in the space. Various examples are available online at:

http://www.funbrain.com/tictactoe/index.html

A version of Tic Tac Toe on a torus is available to play online at:

http://www.math.ntnu.no/~dundas/75060/TorusGames/html/TicTacToe.html

Play Tic Tac Toe over six dimensions! Instructions and game board are available at:

http://drgeorge.org/index.php?area=index&page=game&game=0002

Math Bluff

Introduction

Math Bluff is a game played as a review for the students. It is a great way to practice any topic right before a test. Math

Bluff is fun and it gets the whole class involved.

How to Play

1. Split the class into two teams.

2. Ask Team 1 a question and whoever knows the answer stands up. Students can stand and not know the answer to try and

get more points for their team. That is where the bluff comes in to play. However the other team has a chance to call out a

bluff.

3. Team 2 then picks a student standing on Team 1 to answer the question asked. They would try and pick on someone who

they think would not know the answer.

If that student gets it right you count the amount of people standing and that’s how many points Team 1

would get.

If the student gets the question wrong on Team 1 the student on Team 2 who bluffed Team 1 has the

opportunity to answer the question to get the points.

If that student on Team 2 gets it wrong it goes back to Team 1 and it will keep going back and forth until

someone gets it right or until everyone has had the opportunity to answer it.

If the students struggle with the problem go over it on the board.

4. Ask Team 2 a question and repeat the same process.

5. When a student that is standing answers a question wrong they sit down and the amount of points you get decreases by 1.

6. Try to move quickly to ask as many questions as possible.

7. The team with the most points at the end of the period wins the game.

Connect 4

On the board I have drawn a 5 by 5 grid. I’ve written a problem where students have to solve for x in each square and

have covered each square with a piece of card so the question can’t be seen. I divide the class into 2 or 3 teams.

Students take turns removing a card and answering the question. If they get it correct they put an X across it. Each team

has a different symbol. If the person gets the answer wrong the questions gets covered again and play goes on to the

next team. The first team to connect 4 correct answers in a row, column or diagonally wins.