The Magnetic Tower of Hanoi 1 The Classical

The Magnetic Tower of Hanoi 1

The Classical Tower of Hanoi 2

Classical To. H – video-clip 1 (0: 53) Click link to play a You. Tube video 1. http: //www. youtube. com/watch? v=EHtk 7 k. Zqo. VY 3

Positional Notations 4

Babylonian mathematics (3 -rd millennium BC) It originated with the ancient Sumerians in the 3 rd millennium BC , was transmitted to the Babylonians , and is still used - in modified form - for measuring time , angles , and geographic coordinates. http: //en. wikipedia. org/wiki/Base_60 5

A short reminder of bases (“Positional Notation”) Positional notation From Wikipedia, the free encyclopedia Indian mathematicians developed the Hindu-Arabic numeral system , the modern decimal positional notation, in the 9 -th century. 2506 = 2 x 103 + 5 x 102 + 0 x 101 + 6 x 100 [ Number = SUM npos. *Base ] (Pos. – 1) http: //en. wikipedia. org/wiki/Place_value_system 6

A short reminder of bases – base 2 Base 2 Weight 24 23 22 21 20 Position 5 4 3 2 1 Number 1 0 1 10111(2) = 1*24 +0*23 +1*22 +1*21 +1*20 10111(2) = 16(10) + 4(10) + 2(10) + 1(10) = 23(10) http: //en. wikipedia. org/wiki/Place_value_system 7

A short reminder of bases – base 3 Base 3 Weight 34 33 32 31 30 Position 5 4 3 2 1 Number 1 0 1 2 2 10122(3) = 1*34 +0*33 +1*32 +2*31 +2*30 10122(3) = 81(10) + 9(10) + 6(10) + 2(10) = 98(10) http: //en. wikipedia. org/wiki/Place_value_system 8

The Classical Tower and base 2 9

A. The Classical Tower of Hanoi [To. H] k=N k=2 k=1 A model set of the Towers of Hanoi (with 8 disks) The classical Tower of Hanoi "puzzle" or "mathematical game" invented by the French mathematician Edouard Lucas in 1883. http: //en. wikipedia. org/wiki/Tower_of_Hanoi 10

Classical To. H – video-clip 2 (1: 14) Click link to play a You. Tube video 2. http: //www. youtube. com/watch? v=3 e. GBh. SSxff. M 11

To. H – Puzzle Description Puzzle Components: Three equal posts A set of N different-diameter disks Puzzle-start setting: N disks arranged in a bottom-to-top descending-size order on a "Source" Post Move: Lift a disk off one Post and land it on another Post Disk-placement rules: The Size Rule: A small disk can not "carry" a larger one (Never land a large disk on a smaller one) Puzzle-end state: N disks arranged in a bottom-to-top descending-size order on a "Destination" Post (one of the two originally-free posts) 12

To. H – Recursive Relations 13

To. H – Number of Moves k N 1 2 3 4 5 6 7 1 1 2 3 1 2 4 4 1 2 4 8 5 1 2 4 8 16 6 1 2 4 8 16 32 7 1 2 4 8 16 32 64 8 128 SUM 2 N - 1 1 1 3 3 7 7 15 15 31 31 63 63 127 255 14

Classical To. H – spans “base 2” k N 1 2 3 4 1 1 2 20 1 2 3 20 1 21 2 4 4 20 1 21 2 22 4 8 20 21 22 23 SUM 2 N - 1 1 1 3 21 -1 3 7 22 -1 7 15 23 -1 15 24 -1 15

A “Base 2” game Base 2 Element (k) 1 2 3 4 5 # of moves 1 2 4 8 16 # of moves 20 21 22 23 24 k=N k=2 k=1 16

The Classical Tower Spans base 2 17

Challenge: invent a “base 3” game Can we invent a game that Spans base 3? 18

Elements of a“Base 3” game Base 3 Element (k) 1 2 3 4 5 # of moves 1 3 9 27 81 # of moves 30 31 32 33 34 19

Challenge: invent a “base 3” game So - can we invent a game that Spans base 3? 20

Yes we can 21

MTo. H – video-clip 3 (1: 29) Click link to play a You. Tube video 3. http: //www. youtube. com/watch? v=n. Uo. HHea. J 4 e. I 22

B. The Magnetic Tower of Hanoi [MTo. H] 23

MTo. H – when we where Young and Brave Yaron (10) and a home-made Magnetic Tower of Hanoi Rehovot, Israel - Autumn 1984. 24

MTo. H – Puzzle Description Puzzle Components: Three equal posts. A set of N different-diameter disks Each disk's "bottom" surface is colored Blue and its "top" surface is colored Red Puzzle-start setting: N disks arranged in a bottom-to-top descending-size order on a "Source" Post The Red surface of every disk in the stack is facing upwards Move: Lift a disk off one post Turn the disk upside down and land it on another post Disk-placement rules: ♣The Size Rule: A small disk can not "carry" a larger one (Never land a large disk on a smaller one) ♣The Magnet Rule: Rejection occurs between two equal colors (Never land a disk such that its bottom surface will touch a co-colored top surface of the "resident" disk) Puzzle-end state: N disks arranged in a bottom-to-top descending-size order on a "Destination" Post (one of the two originally-free posts) 25

MTo. H – Solving the N=2 Puzzle 2 1 4 3 26

Colored MTo. H – video-clip 4 (1: 24) Click link to play a You. Tube video 4. http: //www. youtube. com/watch? v=D_xfu. COh 1 S 0 27

B 1. The Colored MTo. H S D I S I D 28

The Colored MTo. H – Number of Moves k N 1 2 3 4 5 6 7 1 1 2 1 3 3 1 3 9 4 1 3 9 27 5 1 3 9 27 81 6 1 3 9 27 81 243 729 8 2187 SUM (3 N - 1)/2 1 1 4 4 13 13 40 40 121 364 1093 3280 29

Colored MTo. H – spans “base 3” k N 1 2 3 4 1 1 2 30 1 3 3 30 1 31 3 9 4 30 2 1 31 2 3 32 2 9 27 30 2 31 2 32 2 33 2 5 SUM (3 N– 1)/2 1 1 (31 -1)/2 4 4 (32 -1)/2 13 13 (33 -1)/2 40 40 (34 -1)/2 30

Challenge met And the fun just begins 31

MTo. H – The Three Versions B 1. The Colored MTo. H B 2. The Semi-Free MTo. H B 3. The Free MTo. H 32

B 2. The Semi-Free MTo. H SS ID DI An MTo. H is Semi-Free if ♣ One of its posts – say – S, is permanently colored – say Red ♣ Another post – say – D, is permanently and oppositely colored ♣ The third post – I - is Free (has a Neutral color at the start of the algorithm) ♣ We need to move N disks from Post S to Post D using Post I 33

The Semi-Free MTo. H – Number of Moves k N 1 k - odd k - even N - odd N - even 2 3 4 5 6 7 1 1 2 1 3 3 1 3 7 4 1 3 7 21 5 1 3 7 21 61 6 1 3 7 21 61 183 7 1 3 7 21 61 183 547 8 SUM 1 4 11 32 93 276 823 1641 2464 34

The Semi-Free MTo. H – Duration Ratio 35

The Free MTo. H • The “ 67” Algorithm • The “ 62” Algorithm 36

The Free MTo. H – video-clip 5 (2: 43) Click link to play a You. Tube video 5. http: //www. youtube. com/watch? v=b. Ztx 5 gexdd. I 37

MTo. H FREEDOM “It is (this) FREEDOM that makes the Magnetic Tower of Hanoi Puzzle so COLORFUL” 38

B 3. The Free MTo. H – The “ 67” Algorithm 39

The “ 67” Algorithm – Number of Moves k N 1 2 3 4 5 6 7 1 1 2 1 3 3 1 3 7 4 1 3 7 19 55 6 1 3 7 19 55 163 7 19 55 163 487 8 1459 SUM 3(N-1) + N-1 1 1 4 4 11 11 30 30 85 85 248 735 2194 40

The “ 67” Algorithm – Duration Ratio 41

B 3. The Free MTo. H – The “ 62” Algorithm 42

The “ 62” Algorithm – Number of Moves k N 1 2 3 4 5 6 7 1 1 2 1 3 3 1 3 7 4 1 3 7 19 53 6 1 3 7 19 53 153 7 19 53 153 455 8 SUM 1 4 11 30 83 236 691 1359 2050 43

The “ 62” Algorithm – Duration Ratio 44

“SF” ; “ 67” ; “ 62” – Duration Ratio 45

“SF” ; “ 67” ; “ 62” – Duration-Ratio Curves 3/4 2/3 67/108 46

The double-pan balance Puzzle 47

Effective (minimum # of) weights for a balance 1 2 3 40 How many? What values? 48

Minimum # of weights - continue 1 2 40 3 27 9 1 3 49

Minimum # of weights - continue 9 1 3 27 1 through 40 9 1 3 81 1 through 121 27 50

Elegance of the “ 67 Algorithm” 51

The “ 67” Algorithm – find a simple rule k N 1 2 3 4 5 6 7 1 2 1 1 3 3 4 5 6 7 8 1 3 7 19 55 163 1 3 7 19 55 163 487 8 SUM 1 4 1459 11 30 85 248 735 2194 52

What about the total number of moves? The “Free 67” Magnetic Tower of Hanoi Total number of moves N SUM 1 1 2 4 3 11 4 30 5 85 53

Recursive Relations 54

Recursive Relations - 1 The “ 100” Algorithm The “ 67” Algorithm 55

Recursive Relations - 2 The “SF” Algorithm k - odd k - even N - odd N - even 56

Recursive Relations - 3 The “ 62” Algorithm k - odd k - even N - odd N - even 57

Recursive Relations - 4 All without exception: 58

Color Crossings 59

MTo. H – Color Crossings - 1 Color of a given post = Red → Neutral → { Red Blue 60

MTo. H – Internet Movie A "movie" showing the "62" Algorithm solving a height five MTo. H in (only) 83 moves: http: //www. numerit. com/maghanoi/ 61

MTo. H – Internet Movie Shown in the movie – solution of the height 5 MTo. H puzzle by (only) 83 moves Click link to play a You. Tube video 6. http: //www. youtube. com/watch? v=sys. N 4 -6 z. XNo It is Freedom that makes the MTo. H so colorful. 62

MTo. H – Color Crossings - 2 The “ 100” Algorithm – NO color crossings 63

MTo. H – Color Crossings - 3 The “ 62” Algorithm – EIGHT color crossings 64

Next 65

“Tower Theory” – Further Modifications Further expansions: ♣ Puzzle-start setting ♣ Number of posts ♣ “Disk" structure (may "quickly" lose its circ. symmetry) ♣ Move rules ♣ Puzzle-end state "Tower Field” in Number Theory? 66

References 67

Gathering 4 Gardner 9 – Atlanta, GA (March `10) 68

Gathering 4 Gardner 9 – Atlanta, GA (March `10) Game inventor: Martin Gardner Figure 6. An artist friend drew this picture for Gardner, illustrating the maximum number of pieces into which a bagel can be sliced by three planes. 69

Gathering 4 Gardner 9 – mini-MTo. H 70

G 4 G 9 - Handouts 71

References [1] "The Magnetic Tower of Hanoi", Uri Levy, Journal of Recreational Mathematics 35: 3, to be published (~May 2010) [2] Paper download: http: //arxiv. org/abs/1003. 0225 [3] "Movie“ (and paper download, different Abstract): http: //www. numerit. com/maghanoi [4] Contact: uri@vicsor. com 72

Cornell University Library http: //arxiv. org/ abs/1003. 0225 73

Realization 74

The Magnetic Tower of Hanoi – Realization 75

The “Colored” Magnetic Tower of Hanoi 76

“Free” or “Classical” MTo. H 77

Oops! 78

Illegal Move! 79

One-Two- Three – GO! 80

The End 81