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(1)

Function Tug of War Tournament

As n

∞

What function is going to win?

(which function will output the larger value)

(2)

New Rule

•

The losing function from the previous

tournament gets a constant helper

•

The constant value can be any number

(3)

Bracket

c (n

2

+ n)

(n

3

-

n

2

)

n

3

+

n

2

c (2n

2

+ 3log(n))

3n

5

+ (log(n))

4

c ((n

4

+ n

2

+ 1) / (n

4

+ 1))

log(n

n

)

c (log(n!))

(4)

Function Tug of War

c (n

2

+ n) --- n

3

– n

2

As n -> ∞

(5)

5

0 2 4 6 8 10 12 14 16 18 20

0 1000 2000 3000 4000 5000 6000 7000 8000

Function Growth Rate: C = 1.25

c(n^2 + n) n^3 - n^2

n

f(

(6)

0 2 4 6 8 10 12 14 16 18 20 0

1000 2000 3000 4000 5000 6000 7000 8000

Function Growth Rate: C = 1.5

c(n^2 + n) n^3 - n^2

n

f(

(7)

7

0 2 4 6 8 10 12 14 16 18 20

0 2000 4000 6000 8000 10000 12000 14000

Function Growth Rate: C = 30

c(n^2 + n) n^3 - n^2

n

f(

(8)

0 5 10 15 20 25 30 35 40 45 50 0

20000 40000 60000 80000 100000 120000 140000

Function Growth Rate: C = 30

c(n^2 + n) n^3 - n^2

n

f(

(9)

9

0 10 20 30 40 50 60 70 80 90 100

0 200000 400000 600000 800000 1000000 1200000

Function Growth Rate: C = 50

c(n^2 + n) n^3 - n^2

n

f(

(10)

Function Tug of War

c (n

2

+ n) --- n

3

– n

2

As n

∞

Who is going to win?

(11)

Function Tug of War

n

3

+

n

2

--- c(

2n

2

+ 3log(n)

)

As n

∞

Who is going to win?

(12)

0 2 4 6 8 10 12 14 16 18 20 0

1000 2000 3000 4000 5000 6000 7000 8000 9000

Function Growth Rate: C = 1.5

c (2n^2 + 3(log(n))) n^3 + n^2

n

f(

(13)

13

0 2 4 6 8 10 12 14 16 18 20

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Function Growth Rate: C = 10

c (2n^2 + 3(log(n))) n^3 + n^2

n

f(

(14)

0 5 10 15 20 25 30 35 40 45 50 0

20000 40000 60000 80000 100000 120000 140000

Function Growth Rate: C = 10

c (2n^2 + 3(log(n))) n^3 + n^2

n

f(

(15)

15

0 10 20 30 40 50 60 70 80 90 100

0 200000 400000 600000 800000 1000000 1200000

Function Growth Rate: C = 30

c (2n^2 + 3(log(n))) n^3 + n^2

n

f(

(16)

Function Tug of War

n

3

+

n

2

--- c(

2n

2

+ 3log(n)

)

As n

∞

Who is going to win?

(17)

Function Tug of War

3n

5

+ (log(n))

4

--- c

((n

4

+ n

2

+ 1) / (n

4

+ 1))

As n

∞

Who is going to win?

(18)

0 1 2 3 4 5 6 7 8 9 10 0

50000 100000 150000 200000 250000 300000 350000

Function Growth Rate: C = 100,000

3n^5 + (log(n))^4 c ((n^4 + n^2 - 1) / (n^4 + 1))

n

f(

(19)

Function Tug of War

3n

5

+ (log(n))

4

--- c

((n

4

+ n

2

+ 1) / (n

4

+ 1))

As n

∞

Who is going to win?

3n

5

+ (log(n))

4
(20)

Function Tug of War

log(n

n

)

--- c(log(n!))

As n

∞

(21)

21

0 2 4 6 8 10 12 14 16 18 20

0 10 20 30 40 50 60 70 80 90 100

Function Growth Rate: C = 1.25

log(n^n) c(log(n!))

n

f(

(22)

0 5 10 15 20 25 0 10 20 30 40 50 60 70 80 90 100

Function Growth Rate: C = 1.4

log(n^n) c(log(n!))

n

f(

(23)

23

0 10 20 30 40 50 60 70 80 90 100

0 100 200 300 400 500 600 700 800

Function Growth Rate: C = 1.4

log(n^n) c(log(n!))

n

f(

(24)

0 2 4 6 8 10 12 14 16 18 20 0

20 40 60 80 100 120 140

Function Growth Rate: C = 2

log(n^n) c(log(n!))

n

f(

(25)

Function Tug of War

log(n

n

)

--- c(log(n!))

As n

∞

Who is going to win?

c(log(n!))

(26)

Bracket

c (n

2

+ n)

n

3

-

n

2

n

3

+

n

2

c (2n

2

+ 3log(n))

3n

5

+ (log(n))

4

c ((n

4

+ n

2

+ 1) / (n

4

+ 1))

log(n

n

)

c (log(n!))

n

3

-

n

2

n

3

+

n

2

3n

5

+ (log(n))

4
(27)

Function Tug of War

c(n

3

-

n

2

)

---

n

3

+

n

2

As n

∞

Who is going to win?

(28)

0 2 4 6 8 10 12 14 16 18 20 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Function Growth Rate: C = 1.25

n^3 + n^2 c(n^3 - n^2)

n

f(

(29)

29

0 2 4 6 8 10 12 14 16 18 20

0 2000 4000 6000 8000 10000 12000

Function Growth Rate: C = 1.5

n^3 + n^2 c(n^3 - n^2)

n

f(

(30)

0 2 4 6 8 10 12 14 16 18 20 0

5000 10000 15000 20000 25000 30000 35000

Function Growth Rate: C = 4

n^3 + n^2 c(n^3 - n^2)

n

f(

(31)

Function Tug of War

c(n

3

-

n

2

)

---

n

3

+

n

2

As n

∞

Who is going to win?

c(n

3

-

n

2

)

(32)

Function Tug of War

3n

5

+ (log(n))

4

---

c(log(n!))

As n

∞

(33)

33

0 2 4 6 8 10 12 14 16 18 20

0 2000000 4000000 6000000 8000000 10000000 12000000

Function Growth Rate: C = 1.25

3n^5 + (log(n))^4 c(log(n!))

n

f(

(34)

0 2 4 6 8 10 12 14 16 18 20 0

2000000 4000000 6000000 8000000 10000000 12000000

Function Growth Rate: C = 500

3n^5 + (log(n))^4 c(log(n!))

n

f(

(35)

35

0 1 2 3 4 5 6 7 8 9 10

0 50000 100000 150000 200000 250000 300000 350000

Function Growth Rate: C = 5000

3n^5 + (log(n))^4 c(log(n!))

n

f(

(36)

Function Tug of War

3n

5

+ (log(n))

4

---

c(log(n!))

As n

∞

Who is going to win?

(37)

Bracket

n

2

+ n

n

3

-

n

2

n

3

+

n

2

2n

2

+ 3log(n)

3n

5

+ (log(n))

4

(n

4

+ n

2

+ 1) / (n

4

+ 1)

log(n

n

)

log(n!)

c(n

3

-

n

2

)

n

3

+

n

2

3n

5

+ (log(n))

4

c(log(n!))

3n

5

+ (log(n))

4

c(n

3

-

n

2

)

(38)

Function Tug of War

c(n

3

-

n

2

)

---

3n

5

+ (log(n))

4

As n

∞

(39)

39

0 2 4 6 8 10 12 14 16 18 20

0 2000000 4000000 6000000 8000000 10000000 12000000

Function Growth Rate: C = 1.25

3n^5 + (log(n))^4 c(n^3 - n^2)

n

f(

(40)

0 2 4 6 8 10 12 14 16 18 20 0

2000000 4000000 6000000 8000000 10000000 12000000

Function Growth Rate: C = 20

3n^5 + (log(n))^4 c(n^3 - n^2)

n

f(

(41)

41

0 2 4 6 8 10 12 14 16 18 20

0 2000000 4000000 6000000 8000000 10000000 12000000

Function Growth Rate: C = 200

3n^5 + (log(n))^4 c(n^3 - n^2)

n

f(

(42)

0 2 4 6 8 10 12 14 16 18 20 0

2000000 4000000 6000000 8000000 10000000 12000000 14000000 16000000

Function Growth Rate: C = 2000

3n^5 + (log(n))^4 c(n^3 - n^2)

n

f(

(43)

43

0 5 10 15 20 25 30 35 40 45 50

0 100000000 200000000 300000000 400000000 500000000 600000000 700000000 800000000 900000000 1000000000

Function Growth Rate: C = 2000

3n^5 + (log(n))^4 c(n^3 - n^2)

n

f(

(44)

Function Tug of War

c(n

3

-

n

2

)

---

3n

5

+ (log(n))

4

As n

∞

Who is going to win?

(45)

Bracket

n

2

+ n

n

3

-

n

2

n

3

+

n

2

2n

2

+ 3log(n)

3n

5

+ (log(n))

4

(n

4

+ n

2

+ 1) / (n

4

+ 1)

log(n

n

)

log(n!)

n

3

-

n

2

n

3

+

n

2

3n

5

+ (log(n))

4

log(n

n

)

3n

5

+ (log(n))

4

n

3

+

n

2

3n

5

+ (log(n))

4

References

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