1. Whiteboardmaths.com
© 2004 All rights reserved 5 7 2 1

2. Edouard Lucas (1884) Probably
The Tower of Hanoi
In the temple of Banares, says he, beneath the dome which marks the centre of the World, rests a brass plate in which are placed 3 diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the creation, god placed 64 discs of pure gold, the largest disc resting on the brass plate and the others getting smaller and smaller up to the top one. This is the tower of brahma. Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of brahma, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it. When the 64 discs shall have been thus transferred from the needle on which at the creation god placed them to one of the other needles, tower, temple and Brahmans alike will crumble into dust and with a thunder clap the world will vanish.
Story
Edouard Lucas (1884) Probably

3. The Tower of Hanoi A B C
5 Tower
Illegal Move

4. The Tower of Hanoi A B C
5 Tower

5. The Tower of Hanoi A B C
3 Tower
Demo 3 tower

6. The Tower of Hanoi
3 Tower A B C

7. The Tower of Hanoi
3 Tower A B C

8. The Tower of Hanoi
3 Tower A B C

9. The Tower of Hanoi
3 Tower A B C

10. The Tower of Hanoi
3 Tower A B C

11. The Tower of Hanoi
3 Tower A B C

12. The Tower of Hanoi
3 Tower
7 Moves A B C

13. The Tower of Hanoi
Confirm that you can move a 3 tower to another peg in a minimum of 7 moves.
Investigate the minimum number of moves required to move different sized towers to another peg.
Try to devise a recording system that helps you keep track of the position of the discs in each tower.
Try to get a feel for how the individual discs move. A good way to start is to learn how to move a 3 tower from any peg to another of your choice in the minimum number of 7 moves.
Record moves for each tower, tabulate results look for patterns make predictions (conjecture) about the minimum number of moves for larger towers, 8, 9, 10,……64 discs. Justification is needed.
How many moves for n disks?
Investigation

14. The Tower of Hanoi A B C
4 Tower
4 Tower show

15. The Tower of Hanoi
4 Tower A B C

16. The Tower of Hanoi
4 Tower A B C

17. The Tower of Hanoi
4 Tower A B C

18. The Tower of Hanoi
4 Tower A B C

19. The Tower of Hanoi
4 Tower A B C

20. The Tower of Hanoi
4 Tower A B C

21. The Tower of Hanoi
4 Tower A B C

22. The Tower of Hanoi
4 Tower A B C

23. The Tower of Hanoi
4 Tower A B C

24. The Tower of Hanoi
4 Tower A B C

25. The Tower of Hanoi
4 Tower A B C

26. The Tower of Hanoi
4 Tower A B C

27. The Tower of Hanoi
4 Tower A B C

28. The Tower of Hanoi
4 Tower A B C

29. The Tower of Hanoi
4 Tower
15 Moves A B C

30. The Tower of Hanoi A B C
5 Tower
5 Tower show

31. The Tower of Hanoi
5 Tower A B C

32. The Tower of Hanoi
5 Tower A B C

33. The Tower of Hanoi
5 Tower A B C

34. The Tower of Hanoi
5 Tower A B C

35. The Tower of Hanoi
5 Tower A B C

36. The Tower of Hanoi
5 Tower A B C

37. The Tower of Hanoi
5 Tower A B C

38. The Tower of Hanoi
5 Tower A B C

39. The Tower of Hanoi
5 Tower A B C

40. The Tower of Hanoi
5 Tower A B C

41. The Tower of Hanoi
5 Tower A B C

42. The Tower of Hanoi
5 Tower A B C

43. The Tower of Hanoi
5 Tower A B C

44. The Tower of Hanoi
5 Tower A B C

45. The Tower of Hanoi
5 Tower A B C

46. The Tower of Hanoi
5 Tower A B C

47. The Tower of Hanoi
5 Tower A B C

48. The Tower of Hanoi
5 Tower A B C

49. The Tower of Hanoi
5 Tower A B C

50. The Tower of Hanoi
5 Tower A B C

51. The Tower of Hanoi
5 Tower A B C

52. The Tower of Hanoi
5 Tower A B C

53. The Tower of Hanoi
5 Tower A B C

54. The Tower of Hanoi
5 Tower A B C

55. The Tower of Hanoi
5 Tower A B C

56. The Tower of Hanoi
5 Tower A B C

57. The Tower of Hanoi
5 Tower A B C

58. The Tower of Hanoi
5 Tower A B C

59. The Tower of Hanoi
5 Tower A B C

60. The Tower of Hanoi
5 Tower A B C

61. The Tower of Hanoi
5 Tower
31 Moves A B C

62. This is called a recursive function.
The Tower of Hanoi
Discs 1
Moves 2 3 4 5 6 7 8 64 n } 1 3
Un = 2Un-1 + 1
This is called a recursive function. 7 15 31 63
127
Why does it happen?
255
Can you find a way to write this indexed number out in full?
264 -1 ?
How long would it take at a rate of 1 disc/second? ?
2n - 1
Results Table

63. Can you use your calculator and knowledge of the laws of indices to work out 264?x
264 = 232 x 232
264 – 1 = 5

64. Moves needed to transfer all 64 discs.
Moves needed to transfer all 64 discs.
Trillions
Billions
Millions
How long would it take if 1 disc/second was moved?
Seconds in a year.
The age of the Universe is currently put at between 15 and years.

65. This is called a recursive function.
The Tower of Hanoi
Un = 2Un-1 + 1
This is called a recursive function. 1 3 7 15 31 63
127
255
Discs
Moves 2 4 5 6 8 n
2n - 1
The proof depends first on proving that the recursive function above is true for all n. Then using a technique called mathematical induction. This is quite a difficult type of proof to learn so I have decided to leave it out. There is nothing stopping you researching it though if you are interested.
We can never be absolutely certain that the minimum number of moves m(n) = 2n – 1 unless we prove it. How do we know for sure that the 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.
Results Table

66. n 5 4 3 2
Regions
Points 1 2 4 8 16
A counter example!
2n-1 6 6 31

67. Historical Note
The Tower of Hanoi was invented by the French mathematician Edouard Lucas and sold as a toy in It originally bore the name of”Prof.Claus” of the college of “Li-Sou-Stain”, but these were soon discovered to be anagrams for “Prof.Lucas” of the college of “Saint Loius”, the university where he worked in Paris.
Edouard Lucas ( )
Lucas studied the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21,… (named after the medieval mathematician, Leonardo of Pisa). Lucas may have been the first person to derive the famous formula for the nth term of this sequence involving the Golden Ratio: … ½(1 + 5).
Lucas also has his own related sequence named after him: 2,1,3,4,7,11,… He went on to devise methods for testing the primality of large numbers and in 1876 he proved that the Mersenne number 2127 – 1 was prime. This remains the largest prime ever found without the aid of a computer.
( )
2127 – 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727
Lucas/Binet formula
Historical Note

68. The King’s Chessboard
According to an old legend King Shirham of India wanted to reward his servant Sissa Ban Dahir for inventing and presenting him with the game of chess. The desire of his servant seemed very modest: “Give me a grain of wheat to put on the first square of this chessboard, and two grains to put on the second square, and four grains to put on the third, and eight grains to put on the fourth and so on, doubling for each successive square, give me enough grain to cover all 64 squares.”
“You don’t ask for much, oh my faithful servant” exclaimed the king. Your wish will certainly be granted.
Based on an extract from “One, Two, Three…Infinity, Dover Publications.
Kings Chessboard

69. How many grains of wheat are on the chessboard?1 2 3 4 5 6 7
nth 1
The King has a problem. 2 4 8 16 32 64
2n-1
The sum of all the grains is: Sn= ………….+ 2n-2 + 2n-1
We need a formula for the sum of this Geometric series.
If Sn= ………….+ 2n-2 + 2n-1
2Sn= ?
………….+ 2n-1 + 2n
2Sn – Sn= ?
2n - 20
Sn= 2n - 1

70. Upper limit of a scientific calculator.
Large numbers
Reading Large Numbers
The numbers given below are the original (British) definitions which are based on powers of a thousand. They are easier to remember however if you write them as powers of a million. They are mostly obsolete these days as the American definitions (smaller) apply in most cases.
Million
Billion*
Trillion
Quadrillion
Quintillion
Sextillion
Septillion
= = 106
= = 1012 (American Trillion)
* The American billion is = and is the one in common usage. A world population of 6.4 billion means
= = 1018
= = 1024
= = 1030
Upper limit of a scientific calculator.
= 1036
= 1042

71. Reading very large numbers
Edouard Lucas ( )
2127 – 1 =
Reading very large numbers
To read a very large number simply section off in groups of 6 from the right and apply Bi, Tri, Quad, Quint, Sext, etc. S Q Q T B M
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.

72. Reading very large numbers
Reading very large numbers
To read a very large number simply section off in groups of 6 from the right and apply Bi, Tri, Quad, Quint, Sext, etc.
Try some of these M B T Q M B T Q M B T Q S M B T Q S

73. Upper limit of a scientific calculator.
How big is a Googol?
Upper limit of a scientific calculator.
1 followed by 100 zeros
The googol was introduced to the world by the American mathematician Edward Kasner ( ). The story goes that when 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 followed by a googol of zeros.
Do we need a number this large? Does it have any physical meaning?
Googol

74. How big is a Googol?
1 followed by 100 zeros
We saw how big 264 was when we converted that many seconds to years:  years. What about a googol of seconds? Who many times bigger is a googol than 264? Use your scientific calculator to get an approximation.
Google

75. Upper limit of a scientific calculator.
How big is a Googol?
Supposing that the Earth was composed solely of the lightest of all atoms (Hydrogen), how many would be contained within the planet?
Earth Mass = 5.98 x 1027 g
The total number of a atoms in the universe has been estimated at 1080.
Hydrogen atom Mass = 1.67 x 10-24g

76. n! is read as n factorial).
Is there a quantity as large as a Googol? 1 2 3 4
Find all possible arrangements for the sets of numbered cards below.
1, 2
4, 2, 3, 1 1
3, 1, 2
4, 3, 1, 2
4, 1, 2, 3
3, 4, 1, 2
1, 4, 2, 3
2, 4, 3, 1 1
2, 1
1, 3, 2
1, 2, 3
3, 1, 4, 2
1, 2, 4, 3
2, 3, 4, 1 2
3, 2, 1
3, 1, 2, 4
1, 2, 3, 4
2, 3, 1, 4
Can you write the number of arrangements as a product of successive integers?
2, 3, 1
4, 1, 3, 2
4, 3, 2, 1
4, 2, 1, 3
2, 1, 3
1, 4, 3, 2
3, 4, 2, 1
2, 4, 1, 3
Objects
arrangements n! 1 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 6
1, 3, 4, 2
3, 2, 4, 1
2, 1, 4, 3
1, 3, 2, 4
3, 2, 1, 4
2, 1, 3, 4
What about if 5 is introduced.Can you see what will happen? 1 2 3 4 5 24
n! is read as n factorial).
Factorials
120

77. Is there a quantity as large as a Googol?
The number of possible arrangements of a set of n objects is given by n! (n factorial). As the number of objects increase the number of arrangements grows very rapidly.
How many arrangements are there for the books on this shelf?
8! =
How many arrangements are there for a suit in a deck of cards?
13! =

78. Is there a quantity as large as a Googol?
The number of possible arrangements of a set of n objects is given by n!.(n factorial) As the number of objects increases the number of arrangements grows very rapidly.
How many arrangements are there for placing the numbers 1 to 16 in the grid? 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1
16! = 2.1 x 1013
How many arrangements are there for the letters of the Alphabet?
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
26! = 4 x 1026

79. 70!  10100 = Googol Is there a quantity as large as a Googol?
The number of possible arrangements of a set of n objects is given by n!.(n factorial) As the number of objects increases the number of arrangements grows very rapidly.
Find other factorial values on your calculator. What is the largest value that the calculator can display?
20!
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
70!  = Googol
So although a googol of physical objects does not exist, if you hold 70 numbered cards in your hand you could theoretically arrange them in a googol number of ways. (An infinite amount of time of course would be needed).

80. What about a Googolplex?
A number so big that it can never be written out in full! There isn’t enough ink,time or paper.
The table shown gives you a feel for how truly unimaginable this number is!
Googolplex

81. And Finally

82. 2000 digits on a page. How many pages needed? The End!
………………….
2000 digits on a page.
How many pages needed?
The End!

83. The Tower of Hanoi
In the temple of Banares, says he, beneath the dome which marks the centre of the World, rests a brass plate in which are placed 3 diamond needles, each a cubit high and as thick as the body of a bee. On one of these needles, at the creation, god placed 64 discs of pure gold, the largest disc resting on the brass plate and the others getting smaller and smaller up to the top one. This is the tower of brahma. Day and night unceasingly the priests transfer the discs from one diamond needle to another according to the fixed and immutable laws of brahma, which require that the priest on duty must not move more than one disc at a time and that he must place this disc on a needle so that there is no smaller disc below it. When the 64 discs shall have been thus transferred from the needle on which at the creation god placed them to one of the other needles, tower, temple and Brahmans alike will crumble into dust and with a thunder clap the world will vanish.
Worksheets

84. A B C
The Tower of Hanoi

86. Tower of Hanoi
Confirm that you can move a 3 tower to another peg in a minimum of 7 moves.
Investigate the minimum number of moves required to move different sized towers to another peg.
Try to devise a recording system that helps you keep track of the position of the discs in each tower.
Try to get a feel for how the individual discs move. A good way to start is to learn how to move a 3 tower from any peg to another of your choice in the minimum number of 7 moves.
Record moves for each tower, tabulate results look for patterns make predictions (conjecture) about the minimum number of moves for larger towers, 8, 9, 10,……64 discs. Justification is needed.
How many moves for n disks?

87. n 5 4 3 2
Regions
Points 1

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