Meeting 4 Student’s Booklet
The 15 Game and other Pastimes
February 8
2017
@ UCI
Contents
1
Tic Tac Toe
STUDENT'S BOOKLET
2 The 15 game
3 Birds and Fish
UC IRVINE MATH CEO
http://www.math.uci.edu/mathceo/
1
TIC TAC TOE
UCI Math CEO
1 Tic Tac Toe
THE GAME OF TIC TAC TOE
• Meeting 4 (FEBRUARY
Team 1: starts and places X’s
Team 2: places O’s
x
First team to complete three
positions forming “a line” wins
(including diagonals, for
example: the line {A, E, I}).
In the game of Tic Tac Toe,
teams take turns placing X’s and O’s
in the positions of a 3 x 3 matrix.
B
C
D
E
F
G
H
I
There are 9 positions in a game of
Tic Tac Toe: A, B, C, D, E, F, G, H and I
x
o
x
o
o
x
Team 1 wins with
the line { A, E, I }
1
A
3
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If you select position H (“bottom middle”), in
how many ways can you win using this position?
Illustrate this by simulating different games.
Draw the games here
1
TIC TAC TOE
UCI Math CEO
2
Play Tic Tac Toe 4 times,
individually or in teams. Make sure
that the starting team plays with X’s.
Record, as a group, all different plays.
Color the starting X move with Green
and the second move with Red. This
will be helpful for analyzing a game.
x
o
x
o
o
x
x
x
o
In this game, the first player put
an X in position H. The second
player put an O in position D.
• Meeting 4 (FEBRUARY
8, 2017)
RECORD YOUR GAMES IN THE NEXT PAGE
USE THIS VOCABULARY WHEN PLAYING::
CENTER
THE POSITION IN THE CENTER
CORNER
THE POSITIONS IN THE CORNERS
MIDDLE
THE POSITIONS THAT ARE NOT
CENTER OF CORNERS
LINES
THE 3 ROWS, 3 COLUMNS AND
2 DIAGONALS
“CHECK!”
ANNOUNCE THAT YOU CAN
POTENTIALLY WIN NEXT TURN
4
TIC TAC TOE GAME TRACKER
GREEN: FIRST MOVE OF THE GAME (X)
RED: SECOND MOVE OF THE GAME (O)
FOR WINNER, WRITE: X, O or TIE
1
WINNER:
5
WINNER:
2
WINNER:
6
WINNER:
3
WINNER:
7
WINNER:
corn
mid
corn
mid
cent
mid
corn
mid
corn
4
WINNER:
8
WINNER:
1
TIC TAC TOE
3 In Tic Tac Toe, you win
by forming a line, which is represented
by a set of 3 positions through
which the line passes.
UCI Math CEO
A B C
D E F
G H I
• Meeting 4 (FEBRUARY
FREQUENCIES
4
The frequency of a position
is the number of lines that pass
through that position.
For example, the diagonal line
drawn here is represented by the set of positions {A, E, I }.
We call {A, E, I} a winning set.
A B C
D E F
G H I
What is the winning set corresponding to the first row?
First row = { ___ , ___ , ___ }
List all the possible lines (winning sets) for Tic Tac Toe:
8, 2017)
You can also think about it as
the number of winning sets
containing that position.
What is the frequency of B?
5
What is the position with
the highest frequency?
6
1
6
TIC TAC TOE
UCI Math CEO
Color all 9 positions as follows:
YELLOW:
RED:
BLUE:
GREEN:
All positions of frequency 1
All positions of frequency 2
All positions of frequency 3
All positions of frequency 4
A B C
D E F
G H I
What kind of symmetry does
this coloring have?
Does it change after a vertical flip?
Does it change after a rotation?
Vertical
Rotate
flip
90
• Meeting 4 (FEBRUARY
8, 2017)
7
7
If you go second in Tic Tac Toe, and you can choose
your opponent’s first move, which “color” position would
you like him to choose? Explain why.
Play a few games, and see if your guess makes you win.
1
TIC TAC TOE
strategies to play
UCI Math CEO
• Meeting 4 (FEBRUARY
o x
Your next move (x):
8
Suppose that you are playing first
(so you put x’s) and the game is
currently at the following stage:
o
x
Is there a strategy you would follow
to lead you to victory, no matter how
your opponent plays? If so, describe
this way. Notice that you are
considering several ways in which
your opponent may play
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How your opponent
may play:
o x
o x
o x
Your next move
o x
o x
o x
8
1
TIC TAC TOE
UCI Math CEO
9
Suppose that you are playing first (so
you put x’s) and the game is currently at
the following stage:
o
x
Play
the
game with
your friend
and try to
win.
• Meeting 4 (FEBRUARY
8, 2017)
9
Describe your strategy.
Your next move (x):
o
x
What happens after:
Is there a strategy that can make you win,
no matter how your opponent plays?
o
x
Play again
and
test
your
strategy.
10
Do you have a strategy to never lose (so: win or
tie) in Tic Tac Toe, even if you play second?
Test your strategy with your peers!
1
TIC TAC TOE
UCI Math CEO
board transformations
Instructions
• Meeting 4 (FEBRUARY
8, 2017)
Vertical
Rotate
flip
90
10
1)
II
Find the transformations of each board
under rotation and vertical flip. Three examples
are shown. Make sure to color the boards.
2)
Questions
1
From boards 1 through 4, Is there one
which “does not change the coloring” no matter
how we transform it?
3)
0 0 1
2
Does this board change coloring if we do a
horizontal flip?
3
How does this board relate to the
frequencies studied before? Explain
1 0 1
1 0 1
4)
0 1 0
1 3 1
0 1 0
1 0 1
0 0 0
1 1 1
2
THE 15 GAME
UCI Math CEO
2 The 15 game
5
7
• Meeting 4 (FEBRUARY 8
1
3
8
6
For example: In this game, team 2 has collected
the numbers 8, 3 and 4. They can say they won,
because the numbers 8, 3 and 4 add to 15:
8
3
4
8+3+4 = 15.
The triplet 8,3,4 is called a winning triplet.
2
Team 1
Team 2
Move 1
6
Move 2
6
4
9
The 15-game is a game played by two teams.
There are nine available numbers (1 through 9), for
example, face up cards. Teams, in turns, pick one
number from the pile. The first team to have three
numbers whose sum is equal to 15 wins the game.
If, after all nine numbers are chosen, no team can
win, the game ends in a tie.
12
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Move 3
2
Move 4
6 2
8
9
8
3
8
3
Move 5
Move 6
4
winning tripiet
2
THE 15 GAME
UCI Math CEO
Notes: It is okay to collect more than three
numbers over the course of the game. For
example, you may collect four or five numbers
but you can only use three of them to sum to 15.
Team 1
Move 4
6
4
6
4
8
4
8
1
winning
have
value:
Move 5
Move 6
5 3
5
3
7
Move 3
3
5 3
13
Because 3+5+7 = 15, the triplet 3,5,7 is a
winning triplet.
Move 1
Move 2
2017)
In this game team 1 has collected the
numbers 5, 3, 6 and 7. They can say they
won, by showing the numbers 3, 5 and 7
(order does not matter):
Team 2
5
5
• Meeting 4 (FEBRUARY 8
7
winning tripiet
Move 7
2
Find
triplets
one
two
different
that
common
2
THE 15 GAME
UCI Math CEO
LET’S PLAY the 15 game!
• Meeting 4 (FEBRUARY
5 7
3
1
8
14
8, 2017)
6
2
4
9
Play this game 4 times. Record the winning triplets
from all the games in which there was a winner. Note: Stop drawing numbers as soon as a team wins.
2
Team 1
Team 2
Team 1
Team 2
Move 1
Move 1
Move 2
Move 2
Move 3
Move 3
Move 4
Move 4
Move 5
Move 5
Move 6
Move 6
Move 7
Move 7
Move 8
Move 8
Move 9
Move 9
2
THE 15 GAME
UCI Math CEO
Team 1
Team 2
• Meeting 4 (FEBRUARY
5 7
3
1
8
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6
9
2
15
4
Move 1
Move 2
Team 1
Team 2
Move 3
Move 1
Move 4
Move 2
Move 5
Move 3
Move 6
Move 4
Move 7
Move 5
Move 8
Move 6
Move 9
Move 7
Move 8
Move 9
2
5
THE 15 GAME
7
3
1
UCI Math CEO
6
8
9
2
3
Play this game 2 more
times but now, right after one
team starts the game, the other
team must write the possible
ways in which the opposite
team can win.
Example: if Team 1 picked a 3,
then Team 2 must calculate all
ways in which Team 1 can win
using 3:
3 + _____ + ______ = 15:
Two possibilities: 3+5+7=15
and 3+4+8=15.
4
• Meeting 4 (FEBRUARY
Team 1
Team 2
Move 1
Move 2
Move 3
Move 4
Move 5
Move 6
Move 7
Move 8
Move 9
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16
2
THE 15 GAME
UCI Math CEO
Team 1
• Meeting 4 (FEBRUARY
Team 2
7
5
Move 1
6
8
9
2
1
3
Move 2
Move 3
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4
Move 4
Move 5
Move 6
4
Based on your experience,
if you are the starting team,
which number is the best to
choose first?
Move 7
Move 8
Move 9
Which
numbers
not
convenient
choose first?
Explain your answer
are
to
2
THE 15 GAME
UCI Math CEO
Example: 3 is contained in two winning
triplers: {3, 5, 7} and {3, 4, 8}, so it has
frequency 2.
5
Find
the
all
winning
triplets.
Then find the frequency of each value.
1: _____
2: ______
3: ______
4: _____
5: ______
6: ______
7: _____
8: ______
9: ______
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MAGIC SQUARES
FREQUENCIES
Choose a value from 1 to 9. The frequency of
that value is equal to the number of winning
triples that contain that value.
• Meeting 4 (FEBRUARY
A magic square is a
square grid such that
all rows, columns and
the two diagonals add
to the same value.
1
Verify that this square is a
magic square by showing that
all lines add to the same value.
8
18
4
6
10
14
16
2
12
2
THE 15 GAME
UCI Math CEO
2
In a magic square, all lines add to the same value. If we
want to place values from 1 through 9 to obtain a magic
square, what should be the sum of each “line”?
Hints: First, find the total sum
1+ 2 + 3 + 4 + 5 + 6+ 7 + 8 + 9 =
The numbers 1 to 9 are placed on three rows of the magic
square, in a way that each row has the same sum.
What is the sum of the value on each row?
What is the sum of the value on each line?
3
When we played the 15 game, we learned that there are
exactly 2 winning triplets containing the value 3.
Color all the
positions of the
magic square
that may
contain a 3.
• Meeting 4 (FEBRUARY
8, 2017)
19
4
Can you place the values 1
through 9 in the square to obtain a
magic square? Try!
1 2 3
4 5 6
7 8 9
If you perform
symmetry on a
magic square,
do you get
another magic
square? Try...
Vertical
flip
Rotate
90
2
THE 15 GAME
UCI Math CEO
• Meeting 4 (FEBRUARY
the 15 game in the magic square board
4
7
5
3
6
1
8
Play the 15 Game again, by using the
value 1 through 9 contained in the magic
square. Any time you pick a number, you color
the corresponding box of the magic square.
TEAM 1: RED
TEAM 2: GREEN
Is there a winning
strategy in the 15 game?
Why?
6
5
1
2
1
9
7
8
3
2
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9
4
What do you notice?
2
3
BIRDS AND FISH
UCI Math CEO
3 Birds and Fish
THE BIRDS AND FISH CARDS
• Meeting 4 (FEBRUARY 8
2017)
22
2
A card is called balanced if it has the
same number of birds as fish. What fraction of
the total number of cards are balanced?
In the game that we will soon consider, there are
different cards. Each card has both birds and fish:
between 1 and 3 birds and between 1 and 3 fish. For
example, the following card has 5 animals:
3
Divide the cards that you have into
categories, according to different properties
relative to the number of objects in the card.
Create different ones. Be creative!
1
Create all possible cards for this
game.
Draw
them
in
a
separate
blank paper. How many did you get?
3
BIRDS AND FISH
UCI Math CEO
• Meeting 4 (FEBRUARY 8
2017)
THE BIRDS AND FISH GAME
Teams take turns picking cards. A team wins if
he fulfills one of the following missions:
M1 Having three cards with the same number
of birds
M2 Having three cards with the same number
of fish
M3 Having three cards with the same total
number of animals
M4 Having three cards such that each card is
balanced (a card is balanced if it has the same
number of birds as fish)
Play this game 4 times!
Example: these three cards would make you win,
as all have the same number of fish (Mission 2)
23
3
BIRDS AND FISH
UCI Math CEO
• Meeting 4 (FEBRUARY 8
2017)
24
PUTTING IT ALL TOGETHER
How would you arrange the 9 cards in a 3x3 grid so that
when playing the game of Fish and Birds, it just feels like
playing Tic Tac Toe? Go for it!
Conclusion or the Day: Tic Tac
Toe, the 15 game and the
game of Fish and Birds are all
mathematically the same!