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

3. Story 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 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.

4. The Tower of Hanoi ABC 5 Tower Illegal Move

5. The Tower of Hanoi ABC 5 Tower

6. Demo 3 tower The Tower of Hanoi ABC 3 Tower

7. The Tower of Hanoi ABC 3 Tower

8. The Tower of Hanoi ABC 3 Tower

9. The Tower of Hanoi ABC 3 Tower

10. The Tower of Hanoi ABC 3 Tower

11. The Tower of Hanoi ABC 3 Tower

12. The Tower of Hanoi ABC 3 Tower

13. The Tower of Hanoi ABC 3 Tower 7 Moves

14. 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

15. 4 Tower show The Tower of Hanoi ABC 4 Tower

16. The Tower of Hanoi ABC 4 Tower

17. The Tower of Hanoi ABC 4 Tower

18. The Tower of Hanoi ABC 4 Tower

19. The Tower of Hanoi ABC 4 Tower

20. The Tower of Hanoi ABC 4 Tower

21. The Tower of Hanoi ABC 4 Tower

22. The Tower of Hanoi ABC 4 Tower

23. The Tower of Hanoi ABC 4 Tower

24. The Tower of Hanoi ABC 4 Tower

25. The Tower of Hanoi ABC 4 Tower

26. The Tower of Hanoi ABC 4 Tower

27. The Tower of Hanoi ABC 4 Tower

28. The Tower of Hanoi ABC 4 Tower

29. The Tower of Hanoi ABC 4 Tower

30. The Tower of Hanoi ABC 4 Tower 15 Moves

31. 5 Tower show The Tower of Hanoi ABC 5 Tower

32. The Tower of Hanoi ABC 5 Tower

33. The Tower of Hanoi ABC 5 Tower

34. The Tower of Hanoi ABC 5 Tower

35. The Tower of Hanoi ABC 5 Tower

36. The Tower of Hanoi ABC 5 Tower

37. The Tower of Hanoi ABC 5 Tower

38. The Tower of Hanoi ABC 5 Tower

39. The Tower of Hanoi ABC 5 Tower

40. The Tower of Hanoi ABC 5 Tower

41. The Tower of Hanoi ABC 5 Tower

42. The Tower of Hanoi ABC 5 Tower

43. The Tower of Hanoi ABC 5 Tower

44. The Tower of Hanoi ABC 5 Tower

45. The Tower of Hanoi ABC 5 Tower

46. The Tower of Hanoi ABC 5 Tower

47. The Tower of Hanoi ABC 5 Tower

48. The Tower of Hanoi ABC 5 Tower

49. The Tower of Hanoi ABC 5 Tower

50. The Tower of Hanoi ABC 5 Tower

51. The Tower of Hanoi ABC 5 Tower

52. The Tower of Hanoi ABC 5 Tower

53. The Tower of Hanoi ABC 5 Tower

54. The Tower of Hanoi ABC 5 Tower

55. The Tower of Hanoi ABC 5 Tower

56. The Tower of Hanoi ABC 5 Tower

57. The Tower of Hanoi ABC 5 Tower

58. The Tower of Hanoi ABC 5 Tower

59. The Tower of Hanoi ABC 5 Tower

60. The Tower of Hanoi ABC 5 Tower

61. The Tower of Hanoi ABC 5 Tower

62. The Tower of Hanoi ABC 5 Tower 31 Moves

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

64. Can you use your calculator and knowledge of the laws of indices to work out 2 64 ? 2 64 = 2 32 x 2 32 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 4294967296 1 8 4 4 6 7 4 4 0 7 3 7 0 9 5 5 1 6 1 6 2 64 – 1 = 5

65. Millions Billions Trillions 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? The age of the Universe is currently put at between 15 and 20 000 000 000 years. Seconds in a year.

66. Results Table The Tower of Hanoi U n = 2U n-1 + 1 This is called a recursive function. 1 3 7 15 31 63 127 255 Discs 1 Moves 2 3 4 5 6 7 8 n2 n - 1 We can never be absolutely certain that the minimum number of moves m(n) = 2 n – 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. 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.

67. n 5 4 3 2 RegionsPoints 1 2 3 4 5 2 4 8 16 6 631 2 n-1 A counter example!

68. Histori cal Note The Tower of Hanoi was invented by the French mathematician Edouard Lucas and sold as a toy in 1883. 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 (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 have been the first person to derive the famous formula for the nth term of this sequence involving the Golden Ratio: 1.61803… 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 2 127 – 1 was prime. This remains the largest prime ever found without the aid of a computer. (1180-1250) 2 127 – 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727 Lucas/Binet formula

69. Kings 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. The King’s Chessboard

70. 1 2 4 8 16 32 64 2 n-1 1 2 3 4 5 6 7 n th How many grains of wheat are on the chessboard? The sum of all the grains is: S n = 2 0 + 2 1 + 2 2 + 2 3 + ………….+ 2 n-2 + 2 n-1 We need a formula for the sum of this Geometric series. If S n = 2 0 + 2 1 + 2 2 + 2 3 + ………….+ 2 n-2 + 2 n-1 2S n = ?2 1 + 2 2 + 2 3 + 2 4 + ………….+ 2 n-1 + 2 n 2S n – S n = ? 2 n - 2 0 S n = 2 n - 1 2 64 - 1 The King has a problem.

71. 1 000 000 1 = 1 000 000 = 10 6 1 000 000 3 = 1 000 000 000 000 000 000 = 10 18 1 000 000 4 = 1 000 000 000 000 000 000 000 000 = 10 24 1 000 000 5 = 1 000 000 000 000 000 000 000 000 000 000 = 10 30 1 000 000 6 = 10 36 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 1 000 000 7 = 10 42 1 000 000 2 = 1000 000 000 000 = 10 12 (American Trillion) * The American billion is = 1 000 000 000 and is the one in common usage. A world population of 6.4 billion means 6 400 000 000. Upper limit of a scientific calculator. Large numbers

72. M BTQQ S 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) 2 127 – 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 off in groups of 6 from the right and apply Bi, Tri, Quad, Quint, Sext, etc.

73. 41 183 460 385 231 191 687 317 716 884 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 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 M B T Q M B T Q Q M B T Q Q SM B T Q Q S S

74. Upper limit of a scientific calculator. How big is a Googol? 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000. 1 followed by 100 zeros Googol The googol was introduced to the world by the American mathematician Edward Kasner (1878-1955). 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 10 googol, that is 1 followed by a googol of zeros. Do we need a number this large? Does it have any physical meaning?

75. How big is a Googol? 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000. 1 followed by 100 zeros Google We saw how big 2 64 was when we converted that many seconds to years:  585 000 000 000 years. What about a googol of seconds? Who many times bigger is a googol than 2 64 ? Use your scientific calculator to get an approximation.

76. Earth Mass = 5.98 x 10 27 g Hydrogen atom Mass = 1.67 x 10 -24 g Upper limit of a scientific calculator. How big is a Googol? 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000. Supposing that the Earth was composed solely of the lightest of all atoms (Hydrogen), how many would be contained within the planet? The total number of a atoms in the universe has been estimated at 10 80.

77. Is there a quantity as large as a Googol? 11 2 3 1 2 34 1 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, 1 2, 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 24 What about if 5 is introduced.Can you see what will happen? 1 2 345 120 Can you write the number of arrangements as a product of successive integers? Objectsarrangements n! 111 222 x 1 363 x 2 x 1 4244 x 3 x 2 x 1 51205 x 4 x 3 x 2 x 1 n! is read as n factorial). Factorials

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 increase the number of arrangements grows very rapidly. How many arrangements are there for the books on this shelf? 8! = 40 320 How many arrangements are there for a suit in a deck of cards? 13! = 6 227 020 800

79. 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. 26! = 4 x 10 26 16! = 2.1 x 10 13 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 How many arrangements are there for placing the numbers 1 to 16 in the grid? 163 213 5 10 118 9 6 7 12 415 14 1

80. Find other factorial values on your calculator. What is the largest value that the calculator can display? 70!  10 100 = Googol 20!2.4 x 10 18 30!2.7 x 10 32 40!8.2 x 10 47 50!3.0 x 10 64 60!8.3 x 10 81 69!1.7 x 10 98 70!Error 52!8.1 x 10 67 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). 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.

81. The table shown gives you a feel for how truly unimaginable this number is! 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. Googolplex

82. And Finally

83. 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000000000000000000…………………. 2000 digits on a page. How many pages needed? The End!

84. 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

85. ABC The Tower of Hanoi

87. 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? Tower of Hanoi

88. n 5 4 3 2 RegionsPoints 1 2 3 4 5

89. 41 183 460 385 231 191 687 317 716 884 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 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