12121212 tower of hanoi math fun 12121212

8/9/2019 12121212 Tower of Hanoi Math Fun 12121212

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The Tower of Hanoi

Edouard Lucas (1884) Probably

In the temple of Banares, says he, beneath the dome which

marks the centre of the World, rests a brass plate in whichare placed 3 diamond needles, each a cubit high and as thickas the body of a bee. On one of these needles, at the

creation, god placed 64 discs of pure gold, the largest discresting on the brass plate and the others getting smaller andsmaller up to the top one. This is the tower of brahma. Day

and night unceasingly the priests transfer the discs from onediamond needle to another according to the fixed and

immutable laws of brahma, which require that the priest onduty must not move more than one disc at a time and that hemust place this disc on a needle so that there is no smaller

disc below it. When the 64 discs shall have been thustransferred from the needle on which at the creation god

placed them to one of the other needles, tower, temple andBrahmans alike will crumble into dust and with a thunder clap

the world will vanish.


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The Tower ofHanoi

A B C

5 Tower

Illegal Move


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The Tower of

Hanoi

A B C

5 Tower


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Demo 3

tower

The Tower of

Hanoi

A B C

3 Tower


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The Tower of

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A B C

3 Tower


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The Tower of

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A B C

3 Tower


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The Tower of

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A B C

3 Tower


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The Tower of

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3 Tower


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The Tower of

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3 Tower


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The Tower of

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A B C

3 Tower


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The Tower of

Hanoi

A B C

3 Tower

7 Moves


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The Tower of HanoiConfirm that you can move a 3 tower to another peg in a minimumof 7 moves.

Investigate the minimum number of moves required to movedifferent sized towers to another peg.

Try to devise a recording system that helps you keep track ofthe position of the discs in each tower.

Try to get a feel for how the individual discs move. A good way tostart is to learn how to move a 3 tower from any peg to anotherof your choice in the minimum number of 7 moves.

Record moves for each tower, tabulate results look for patternsmake predictions (conjecture) about the minimum number of

moves for larger towers, 8, 9, 10,64 discs. Justification isneeded.

How many moves for n disks? Investigation


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4 Tower show

The Tower of

Hanoi

A B C

4 Tower


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4 Tower


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4 Tower


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4 Tower


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The Tower of

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4 Tower

15 Moves


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5 Tower show

The Tower of

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5 Tower


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The Tower of

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A B C

5 Tower

31 Moves

The Tower of Hanoi


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13

7

15

3163

127

255

Results Table

?

The Tower of Hanoi

Discs

1

Moves

2

3

4

56

7

8

64

n ?

Un = 2Un-1 + 1

This is called a

recursive function.

2n

- 1

264 -1

Why does it happen?

How long would it take at a

rate of 1 disc/second?

Can you find a way to writethis indexed number out infull?


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Can you use your calculator and knowledge of thelaws of indices to work out 264?

264 = 232 x 232

2 5 7 6 9 8 0 3 7 7 6

3 8 6 5 4 7 0 5 6 6 4 0

8 5 8 9 9 3 4 5 9 2 0 0

3 0 0 6 4 7 7 1 0 7 2 0 0 0

2 5 7 6 9 8 0 3 7 7 6 0 0 0 0

3 8 6 5 4 7 0 5 6 6 4 0 0 0 0 0

1 7 1 7 9 8 6 9 1 8 4 0 0 0 0 0 0

3 8 6 5 4 7 0 5 6 6 4 0 0 0 0 0 0 0

8 5 8 9 9 3 4 5 9 2 0 0 0 0 0 0 0 0

1 7 1 7 9 8 6 9 1 8 4 0 0 0 0 0 0 0 0 0

x 42949672964294967296

1 8 4 4 6 7 4 4 0 7 3 7 0 9 5 5 1 6 1 6264 1 = 5


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MillionsBillionsTrillions

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

Moves needed to transfer all 64 discs.

How long would it take if 1 disc/second was moved?

585 000 000 000 years}

The age of the Universe is currently put at between15 and 20 000 000 000 years.

64112 1 5.85 10 y rs

(60 60 24 365)x

x x x

Seconds in a year.

The Tower of Hanoi


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Results Table

The Tower of Hanoi

Un = 2Un-1 + 1

This is called arecursive function.

13

7

15

3163

127

255

Discs

1

Moves

2

3

4

56

7

8

n 2n - 1

We can never be absolutely certain that the minimum number ofmoves m(n) = 2n 1 unless we prove it. How do we know for sure thatthe rule will not fail at some future value of n? If it did then this

would be a counter example to the rule and would disprove it.

The proof depends first onproving that the recursivefunction above is true for all n.

Then using a technique calledmathematical induction. This isquite a difficult type of proof tolearn so I have decided to leave itout. There is nothing stopping you

researching it though if you areinterested.


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n

54

3

2

RegionsPoints1

2

3

45

2

4

816

66 31

2n-1

A counter example!

Historical Note


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Histori

cal

Note

Historical Note

The Tower of Hanoi was invented by the Frenchmathematician Edouard Lucas and sold as a toy in 1883.

It originally bore the name ofProf.C

laus of the collegeof Li-Sou-Stain, but these were soon discovered to beanagrams for Prof.Lucas of the college of SaintLoius, the university where he worked in Paris.

Edouard Lucas(1842-1891)

Lucas studied the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, (named

after the medieval mathematician, Leonardo of Pisa). Lucas may havebeen the first person to derive the famous formula for the nth termof this sequence involving the Golden Ratio: 1.61803 (1 + 5).

Lucas also has his own related sequence named after him: 2,1,3,4,7,11, Hewent on to devise methods for testing the primality of large numbers andin 1876 he proved that the Mersenne number 2127 1 was prime. Thisremains the largest prime ever found without the aid of a computer.

(1180-1250)(1 5) (1 5)

5

n n

n nF

!

2127

1 = 170,141,183,460,469,231,731,687,303,715,884,105,727

Lucas/Binetformula

h h


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Kings

Chessboard

According to an old legend King Shirham of India wanted to

reward his servant Sissa Ban Dahir for inventing and presentinghim with the game of chess. The desire of his servant seemedvery modest: Give me a grain of wheat to put on the firstsquare of this chessboard, and two grains to put on the secondsquare, and four grains to put on the third, and eight grains to

put on the fourth and so on, doubling for each successivesquare, give me enough grain to cover all 64 squares.

You dont ask for much, oh my faithful servant exclaimed theking. Your wish will certainly be granted.

Based on an extract from One, Two, ThreeInfinity, Dover Publications.

The Kings Chessboard

How many grains of wheat are on the chessboard?


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1

24

8

16

32

64

2n-1

1

23

4

5

6

7

nth

How many grains of wheat are on the chessboard?

The sum of all the grains is:S

n= 20

+ 21

+ 22

+ 23

+ .+ 2n-2

+ 2n-1

We need a formula for the sum of this Geometric series.

If Sn= 20 + 21 + 22 + 23 + .+ 2n-2 + 2n-1

2Sn= ?21 + 22 + 23 + 24 + .+ 2n-1 + 2n

2Sn Sn= ?2n - 20

Sn= 2n - 1

264 - 1

The King hasa problem.


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70/87

MBTQQS

One hundred and seventy sextillion,

one hundred and forty one thousand, one hundred and eighty three quintillion,

four hundred and sixty thousand, four hundred and sixty nine quadrillion,two hundred and thirty one thousand, seven hundred and thirty one trillion,

six hundred and eighty seven thousand, three hundred and three billion,

seven hundred and fifteen thousand, eight hundred and eighty four million,

one hundred and five thousand, seven hundred and twenty seven.

Edouard Lucas(1842-1891)

2127

1 = 170 141 183 460 469 231 731 687 303 715 884 105 727

Reading very large numbers

To read a very large number simply section offin groups of 6 from the right and apply Bi, Tri,Quad, Quint, Sext, etc.


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41 183 460 385 231 191 687 317 716 884


To read a very large number simply section off in groups of 6from the right and apply Bi, Tri, Quad, Quint, Sext, etc.

Try some of these

57 786 765 432 167 876 564 875 432 897 675 432

9 412 675 987 453 256 645 321 786 765 786 444 329 576

678 876 543 786 543 987 579 953 237 896 764 345 675 876 453 231

MBTQ

MBTQQ

MBTQQS

MBTQQSS


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10010Googol !Upper limit

of ascientificcalculator.

How big is a Googol?

10 000 000 000 000 000 000 000 000 000 000 000 000 000000 000 000 000 000 000 000 000 000 000 000 000 000 000

000 000 000 000 000 000. 1 followed by 100 zeros

Google

The googol was introduced to the world by the Americanmathematician Edward Kasner (1878-1955). The story goes thatwhen he asked his 8 year old nephew, Milton, what name he

would like to give to a really large number, he replied googol.Kasner also defined the Googolplex as 10googol, that is 1 followedby a googol of zeros.

Do we need a number this large? Does it have any physical meaning?


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10010Googol !


10 000 000 000 000 000 000 000 000 000 000 000 000 000000 000 000 000 000 000 000 000 000 000 000 000 000 000000 000 000 000 000 000.

1 followed by 100 zeros

Google

We saw how big 264 was when we converted that many secondsto years: } 585 000 000 000 years. What about a googol ofseconds? Who many times bigger is a googol than 264? Use yourscientific calculator to get an approximation.

10080

64

105.4 10

2x}

80 11

92

5.4 10 5.85 10

3 10 y rs.

So x x x

x

}

}


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Earth Mass = 5.98 x 1027 g

Hydrogen atomMass = 1.67 x 10-24g

10010Googol !Upper limit

of ascientificcalculator.


10 000 000 000 000 000 000 000 000000 000 000 000 000 000 000 000 000000 000 000 000 000 000 000 000 000000 000 000 000 000 000 000.

Supposing that the Earth was composedsolely of the lightest of all atoms(Hydrogen), how many would becontained within the planet?

2751

24

5.98 103.58 10

1.67 10

xx Googol

x ! }

The total number of a atoms in theuniverse has been estimated at 1080.

Is th tit s l s G l?


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Is there a quantity as large as a Googol?

1 1 2 3 1 2 3 41 2

Find all possible arrangements for the sets of numbered cards below.

1 1, 2

2, 1

3, 1, 2

1, 3, 2

1, 2, 3

3, 2, 12, 3, 1

2, 1, 3

4, 3, 1, 2

3, 4, 1, 2

3, 1, 4, 2

3, 1, 2, 4

4, 1, 3, 2

1, 4, 3, 2

1, 3, 4, 2

1, 3, 2, 4

4, 1, 2, 3

1, 4, 2, 3

1, 2, 4, 3

1, 2, 3, 4

4, 3, 2, 1

3, 4, 2, 1

3, 2, 4, 1

3, 2, 1, 4

4, 2, 3, 1

2, 4, 3, 1

2, 3, 4, 1

2, 3, 1, 4

4, 2, 1, 3

2, 4, 1, 3

2, 1, 4, 3

2, 1, 3, 4

1

2

6

24What about if 5 is introduced.Can

you see what will happen?

1 2 3 4 5

120

Can you write the number ofarrangements as a product ofsuccessive integers?

Objects arrangements n!

1 1 1

2 2 2 x 1

3 6 3 x 2 x 1

4 24 4 x 3 x 2 x 1

5 120 5 x 4 x 3 x 2 x 1

n! is read as n factorial).F

actorials

Is there a quantity as lar e as a Goo ol?


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The number of possible arrangements of a set of n objects is given

by n! (n factorial). As the number of objects increase the numberof arrangements grows very rapidly.

How many arrangements are therefor the books on this shelf?

8! = 40 320

How many arrangements are therefor a suit in a deck of cards?

13! = 6 227 020 800



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by n!.(n factorial) As the number of objects increases the numberof arrangements grows very rapidly.

26! = 4 x 1026

16! = 2.1 x 1013

How many arrangements are therefor the letters of the Alphabet?

A BC

D E F G H I J K L M N O P QR

S

T U V W X Y Z

How many arrangements are there forplacing the numbers 1 to 16 in the grid?

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1



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Find other factorial values on your calculator. What is thelargest value that the calculator can display?

70! } 10100 = Googol20! 2.4 x 1018

30! 2.7 x 1032

40! 8.2 x 1047

50! 3.0 x 1064

60! 8.3 x 1081

69! 1.7 x 1098

70! Error

52! 8.1 x 1067

So although a googol of physicalobjects does not exist, if youhold 70 numbered cards in yourhand you could theoreticallyarrange them in a googol numberof ways. (An infinite amount oftime of course would be needed).



by n!.(n factorial) As the number of objects increases the numberof arrangements grows very rapidly.

What about a Googolplex?


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2

3

6

12

18

24

30

10

10

10

10

10

10

10

10 1 with 100 z r s ( l )

10 1 with 1000 z r s

10 1 with 1000000 z r s

10 1 with 1 billi z r s

10 1 with 1 trilli z r s

10 1 with qu drilli z r s

10 1 with

!

!

!

!

!

!

!

36

42

100

10

10

10

a qui tilli z r s

10 1 with a s xtili z r s

10 1 with a s tili z r s

10 1 with a l z r s

!

!

!

The table shown givesyou a feel for howtruly unimaginable thisnumber is!

What about a Googolplex?

10010 10 10googolA Googol l x ! !

A number so big that it cannever be written out in full!There isnt enough ink,timeor paper.

Googolplex


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AndFinally

2000 digits on a page How many pages needed?


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1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000.

2000 digits on a page.

10010 10010 1 10 .A Googolplex follo e y zeros! !

How many pages needed?

100

3

9610

2 10

5 10Pages nee e xd

x

! ! The End!

The Tower of Hanoi


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The Tower of HanoiIn the temple of Banares, says he, beneath the dome whichmarks the centre of the World, rests a brass plate in which

are placed 3 diamond needles, each a cubit high and as thickas the body of a bee. On one of these needles, at the

creation, god placed 64 discs of pure gold, the largest discresting on the brass plate and the others getting smaller andsmaller up to the top one. This is the tower of brahma. Day

and night unceasingly the priests transfer the discs from onediamond needle to another according to the fixed and

immutable laws of brahma, which require that the priest onduty must not move more than one disc at a time and that hemust place this disc on a needle so that there is no smaller

disc below it. When the 64 discs shall have been thustransferred from the needle on which at the creation god

placed them to one of the other needles, tower, temple andBrahmans alike will crumble into dust and with a thunder clap

the world will vanish.Worksheets


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A B C

The Tower of Hanoi


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Confirm that you can move a 3 tower to another peg in a minimumof 7 moves.

Investigate the minimum number of moves required to movedifferent sized towers to another peg.

Try to devise a recording system that helps you keep track ofthe position of the discs in each tower.

Try to get a feel for how the individual discs move. A good way tostart is to learn how to move a 3 tower from any peg to anotherof yourchoice in the minimum number of 7 moves.

Record moves for each tower, tabulate results look for patternsmake predictions (conjecture) about the minimum number ofmoves for larger towers, 8, 9, 10,64 discs. Justification isneeded.

How many moves for n disks?

Tower of Hanoi


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n

5

4

3

2

RegionsPoints1

2

3

45


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41 183 460 385 231 191 687 317 716 884


To read a very large number simply section off in groups of 6from the right and apply Bi, Tri, Quad, Quint, Sext, etc.

Try some of these

57 786 765 432 167 876 564 875 432 897 675 432

9 412 675 987 453 256 645 321 786 765 786 444 329 576

678 876 543 786 543 987 579 953 237 896 764 345 675 876 453 231

12121212 tower of hanoi math fun 12121212

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