Number Games
A Lesson in the “Math + Fun!” Series
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Apr. 2005
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Number Games
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Slide 1
About This Presentation
This presentation is part of the “Math + Fun!” series devised
by Behrooz Parhami, Professor of Computer Engineering at
University of California, Santa Barbara. It was first prepared
for special lessons in mathematics at Goleta Family School
during the 2003-04 and 2004-05 school years. The slides can
be used freely in teaching and in other educational settings.
Unauthorized uses are strictly prohibited. © Behrooz Parhami
Apr. 2005
Edition
Released
First
Apr. 2005
Revised
Number Games
Revised
Slide 2
What Number Did You Choose?
1. Pick a number from this list:
1
2
3
4
5
6
7
8
9 10
2. Multiply the number by 9. If the result has two digits, add the two digits;
otherwise leave the number alone.
3. Subtract 5 from the number that you got in step 2.
4. Pick the letter that corresponds to the number you got in step 3.
1 = A, 2 = B, 3 = C, 4 = D, 5 = E, 6 = F, 7 = G, 8 = H, etc.
5. Pick a country in Europe whose name begins with your letter.
A = Austria, . . .
C = Croatia, . . .
E = England, . . .
B = Belgium, . . .
D = Denmark, . . .
F = France, . . .
etc.
6. Use the last letter in your country’s name as the first letter of an animal.
7. Use the last letter of your animal’s name as the first letter of a color.
Apr. 2005
Number Games
Slide 3
How Did the Game Work?
1. Pick a number from this list:
1
2
3
4
5
6
7
8
9 10
2. Multiply the number by 9.
9 18 27 36 45 54 63 72 81 90
If the result has two digits, add the two digits
You get 9 in all cases!
3. Subtract 5 from the number that you got in step 2.
You get 4
4. Pick the letter that corresponds to the number you got in step 3.
1 = A, 2 = B, 3 = C, 4 = D, 5 = E, 6 = F, 7 = G, 8 = H, etc.
5. Pick a country in Europe whose name begins with your letter.
A = Austria, . . .
C = Croatia, . . .
E = England, . . .
B = Belgium, . . .
D = Denmark, . . .
F = France, . . .
Kangaroo
6. Use the last letter in your country’s name as the first letter of an animal.
Orange
7. Use the last letter of your animal’s name as the first letter of a color.
Apr. 2005
Number Games
Slide 4
More Mind-Reading Games
1.
2.
3.
4.
5.
Think of a 2-digit number (10-99).
Multiply your number by 5.
Add 500 to the result of step 2.
Add the number of your siblings.
Double the result of step 4.
You got a 4-digit number that
starts with 1, has your original
2-digit number in the middle,
and double the number of
your siblings at the end.
1. Think of a number between 1 and 30.
2. Tell me in which of the five lists below the number appears.
List A: 1 3 5 7
List B: 2 3 6 7
List C: 4 5 6 7
List D: 8 9 10 11
List E: 16 17 18 19
Apr. 2005
9
10
12
12
20
11
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14
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15
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20
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19
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25
25
Number Games
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30
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30
A
B
C
E
Slide 5
D
E
Activity 1: Mind-Reading Game
Play the following mind-reading game with your friends and family
(First, discover how to tell the number if you know which lists it is in)
1. Think of a number between 1 and 20.
2. Tell me in which of the five lists below the number appears.
List A: 1 4 6 8
List B: 2 6 7 8
List C: 3 4 6 9
List D: 5 7 8 9
List E: 10 11 12 13
9
12
13
15
14
11
17
14
16
15
14
20
18
17
16
16 19
19 20
18 19 20
17 18 19 20
Challenge: Add a List F to the five lists given above, and complete
the lists, so that you can play the game with numbers up to 40.
Apr. 2005
Number Games
Slide 6
Activity 2: More Mind-Reading Games
I know what your number is!
1.
2.
3.
4.
5.
6.
7.
Pick any number and write it down
Add 10 to your number
Add 20 to the result
Subtract 4 from the result
Add 15 to the result
Subtract 5 from the result
Subtract your original number
Your final result is 36, isn’t it?
Explain how this game works.
Change the numbers to make
your own game.
The Birthday Number Game
1.
2.
3.
4.
5.
6.
7.
8.
Write down your BD month (1-12)
Multiply your number by 5
Add 6 to the result
Multiply the result by 4
Add 9 to the result
Multiply the result by 5
Add the day of your BD (1-31)
Subtract 165 from your result
Apr. 2005
Number Games
Your final result shows the
month and day of your birth
Explain how this game works.
Slide 7
Euclid’s Game for Two Players
A referee writes two numbers on the board
The two players take turns writing the difference of two numbers
already on the board, provided the difference is a new number
The player who cannot write a new number loses
Player 1
loses
Referee
24
Player
1
18
6
Player
2
80
15
65
12
35
50
The total number of values written on the board is
n = the larger number / greatest common divisor of the two numbers
If n is odd, player 1 wins
Apr. 2005
Number Games
Slide 8
Activity 3: The Game of Nim
Write three numbers on three lines
The two players take turns reducing one of the numbers
The player who makes the last move loses
Start
5
3
1
Player 1
2
3
1
Player 2
2
2
1
Player 1
2
2
0
Player 2
2
1
0
Player 1
0
1
0
Player 2
0
0
0
Loses!
Challenge:
1. Discover a winning strategy with (5, 3, 1), if you are the first player
2. Play the game with other numbers such as (7, 4, 2), (6, 5, 3), . . .
Apr. 2005
Number Games
Slide 9
The Four 4s Puzzle
Use four 4s and any number of math symbols (such as +, -,
, /, , . . .) to form as many different numbers as possible.
0=4–4+4–4
Examples
1 = 4 – 4 + 4/4
2 = 4 / 4 + 4 / 4 or 4 – (4 + 4) / 4
3 = (4 + 4 + 4) / 4
4 = (4 + 4) / 4 + 4
5 = (4  4 + 4) / 4
6 = (4 + 4 + 4) / 4
7 = 4 + 4 – 4 / 4 or 44 / 4 – 4
8 = 4  (4 / 4) + 4
9 = 44 / 4 – 4 or 4 / 4 + 4  4
10 = (4 + 4 / 4)  4
11 = 44 / (4 + 4)
12 = 4  (4 – 4 / 4)
Apr. 2005
13 = 44 / 4 + 4
14 = 4  (4 + 4) – 4
15 = 44 / 4 + 4
16 = 4  4 + 4 – 4 or 4  4  4 / 4
17 = 4  4 + 4 / 4
18 = 4  (4 + 4) + 4
19 = 4.4  4.4
20 = 4  (4 + 4 / 4)
21 = 4.4  4 + 4
22 = 4  4 + 4 + 4
23 = 4  (4 + 4) – .4
24 = 4  (4 + 4 + 4)
25 = (4 + 4 / 4)4
Number Games
Slide 10
Activity 4: The Five 5s Puzzle
Use five 5s and any number of math symbols (such as +, -,
, /, , . . .) to form as many different numbers as possible.
0 = 5  (5 / 5 – 5 / 5)
1 = 55 / 5 – 5 – 5
2=
3=
4=
5=
6=
7=
8=
9=
10 =
11 =
12 =
Apr. 2005
Examples
13 =
14 =
15 =
16 =
17 =
18 =
19 =
20 =
21 =
22 =
23 =
24 =
25 =
Number Games
Slide 11
Activity 5: Birth Year Puzzle
Use the digits in your year of birth, in order, plus any number
of math symbols (+, –, . . .) to form the numbers from 0 to 10.
Examples for 1993
0 = 1  (9 – 9)  3
1 = 1 + (9 – 9)  3
2 = –1 + 9 – 9 + 3
3 = (1 + 9 – 9)  3
4=
5=
6 = (1+ 9 / 9)  3
7=
8=
9=
10 =
Apr. 2005
Examples for 1994
0 = 1  (9 – 9)  4
1 = 1 + (9 – 9)  4
2 = –1 – 9 / 9 + 4
3 = –1 + 9 – 9 + 4
4=19–9+4
5=1+9–9+4
6 = 1 + 9/9 + 4
7 = 1  (9 / 9) + 4
8 = (1 + 9 / 9)  4
9 = 1 + 9 + 9 – 4
10 = 1  9 – 9 + 4
Number Games
Examples for 1995
0 = 1  (9 – 9)  5
1 = 1 + (9 – 9)  5
2=
3=
4=
5 = 1 – 9/9 + 5
6=
7=
8=
9=
10 = (1 + 9 / 9)  5
Slide 12
Activity 6: Your Phone Number
Write down your phone number, with space between digits
Example:
5
5
5
1
2
5
3
Now, try to make as many different numbers as you can by
Putting + , - , , / signs between the digits.
0
1
2
3
4
5
6
7
8
Apr. 2005
=
=
=
=
=
=
=
=
=
5 5 +
5 +
5
5
5
5
5
5
5 +
5 +
5 +
5
5
5
5
5
5
5 +
5 5 +
5
5
5
5
5
5
1 +
1 +
1 
1
1
1
1
1
1
Number Games
2 2 2 2
2
2
2
2
2
5 5 
5 
5
5
5
5
5
5
3
3
3
3
3
3
3
3
3
Slide 13
Another Game for Two Players
Put some playing cards (or numbers) in a row
Players take turns removing a card from either end of the row
At the end, add up the numbers to see who wins
Apr. 2005
Number Games
Slide 14
Game Tree
Score for
Player 1
Score for
Player 2
Cards left
0 [1 2 5 4 8 6] 0
Player 1
1 [2 5 4 8 6] 0
6 [1 2 5 4 8] 0
Player 2
1 [5 4 8 6] 2
1 [2 5 4 8] 6
6 [2 5 4 8] 1
6 [1 2 5 4] 8
Player 1
6 [4 8 6] 2
7 [5 4 8] 2
3 [5 4 8] 6
9 [2 5 4] 6
Player 2
6 [8 6] 6
7 [4 8] 7
6 [4 8] 8
Player 1
1
1
3 [4 8] 11
7 [5 4] 10
1
2
9 [5 4] 8
3 [5 4] 14
2
2
9 [2 5] 10
1
1
Player who wins
Apr. 2005
Number Games
Slide 15
Some Surprising Predictions
Answer these questions quickly.
If you pause, it won’t work.
What is 2 + 2?
What is 4 + 4?
What is 1 + 1?
What is 9 + 9?
What is 2 + 2?
What is the
first vegetable
that comes to
your mind?
What is 3 + 3?
What is 4 + 4?
What is 5 + 5?
What is 8 + 8?
What is 16 + 16?
Quickly, pick a
number between
12 and 5
What is 6 + 6?
What is 7 + 7?
No one knows why these guesses
work for most people.
What is 8 + 8?
Apr. 2005
Number Games
Slide 16
Next Lesson
Thursday, May 19, 2005
What is special about 40? (Hint: Forty)
The only number with letters
appearing in alphabetical order
What is special about 6? (Hint: Divisible by 1, 2, 3)
What is special about 38? (Hint: XXXVIII)
Apr. 2005
Number Games
The smallest
perfect number
The last Roman numeral
in alphabetical order
Slide 17