The Magnetic Tower of Hanoi

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The Magnetic Tower of Hanoi
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The Classical Tower of Hanoi
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Classical ToH video-clip 1 (053)
Click link to play a YouTube video 1.
http//www.youtube.com/watch?vEHtk7kZqoVY
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Positional Notations
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Babylonian mathematics (3-rd millennium BC)
It originated with the ancient Sumerians in the
3rd 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
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A short reminder of bases (Positional Notation)
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
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A short reminder of bases base 2
Base 2
20 21 22 23 24 Weight
1 2 3 4 5 Position
1 1 1 0 1 Number
10111(2) 124 023 122 121 120
10111(2) 16(10) 4(10) 2(10) 1(10)
23(10)
http//en.wikipedia.org/wiki/Place_value_system
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A short reminder of bases base 3
Base 3
30 31 32 33 34 Weight
1 2 3 4 5 Position
2 2 1 0 1 Number
10122(3) 134 033 132 231 230
10122(3) 81(10) 9(10) 6(10) 2(10)
98(10)
http//en.wikipedia.org/wiki/Place_value_system
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The Classical Tower and base 2
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A. The Classical Tower of Hanoi ToH
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
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Classical ToH video-clip 2 (114)
Click link to play a YouTube video 2.
http//www.youtube.com/watch?v3eGBhSSxffM
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ToH 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)
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ToH Recursive Relations
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ToH Number of Moves
2N - 1 SUM 8 7 6 5 4 3 2 1 k N
1 1 1 1
3 3 2 1 2
7 7 4 2 1 3
15 15 8 4 2 1 4
31 31 16 8 4 2 1 5
63 63 32 16 8 4 2 1 6
127 127 64 32 16 8 4 2 1 7
255 255 128 64 32 16 8 4 2 1 8
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Classical ToH spans base 2
2N - 1 SUM 4 3 2 1 k N
1 1 1 1

3 3 2 1 2

7 7 4 2 1 3

15 15 8 4 2 1 4

21-1 20
22-1 21 20
22 21 20
23-1 22 21 20
23 22 21 20
24-1 23 22 21 20
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A Base 2 game
Base 2
5 4 3 2 1 Element (k)
16 8 4 2 1 of moves
24 23 22 21 20 of moves
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The Classical Tower Spans base 2
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Challenge invent a base 3 game
Can we invent a game that Spans base 3?
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Elements of aBase 3 game
Base 3
5 4 3 2 1 Element (k)
81 27 9 3 1 of moves
34 33 32 31 30 of moves
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Challenge invent a base 3 game
So - can we invent a game that Spans base 3?
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Yes we can
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MToH video-clip 3 (129)
Click link to play a YouTube video 3.
http//www.youtube.com/watch?vnUoHHeaJ4eI
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B. The Magnetic Tower of Hanoi MToH
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MToH when we where Young and Brave
Yaron (10) and a home-made Magnetic Tower of
Hanoi Rehovot, Israel - Autumn 1984.
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MToH 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)

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MToH Solving the N2 Puzzle
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Colored MToH video-clip 4 (124)
Click link to play a YouTube video 4.
http//www.youtube.com/watch?vD_xfuCOh1S0
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B1. The Colored MToH
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The Colored MToH Number of Moves
(3N - 1)/2 SUM 8 7 6 5 4 3 2 1 k N
1 1 1 1
4 4 3 1 2
13 13 9 3 1 3
40 40 27 9 3 1 4
121 121 81 27 9 3 1 5
364 364 243 81 27 9 3 1 6
1093 1093 729 243 81 27 9 3 1 7
3280 3280 2187 729 243 81 27 9 3 1 8
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Colored MToH spans base 3
(3N1)/2 SUM 5 4 3 2 1 k N
1 1 1 1

4 4 3 1 2

13 13 9 3 1 3

40 40 27 9 3 1 4

(31-1)/2 30
(32-1)/2 31 30
22 21 20
(33-1)/2 32 31 30
23 22 21 20
(34-1)/2 33 32 31 30
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Challenge met
And the fun just begins
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MToH The Three Versions
B1. The Colored MToH B2. The Semi-Free MToH B3.
The Free MToH
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B2. The Semi-Free MToH

• An MToH 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

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The Semi-Free MToH Number of Moves
SUM 8 7 6 5 4 3 2 1 k N
1 1 1
4 3 1 2
11 7 3 1 3
32 21 7 3 1 4
93 61 21 7 3 1 5
276 183 61 21 7 3 1 6
823 547 183 61 21 7 3 1 7
2464 1641 547 183 61 21 7 3 1 8
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The Semi-Free MToH Duration Ratio
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The Free MToH

• The 67 Algorithm
• The 62 Algorithm

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The Free MToH video-clip 5 (243)
Click link to play a YouTube video 5.
http//www.youtube.com/watch?vbZtx5gexddI
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MToH FREEDOM
It is (this) FREEDOM that makes the Magnetic
Tower of Hanoi Puzzle so COLORFUL
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B3. The Free MToH The 67 Algorithm
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The 67 Algorithm Number of Moves
3(N-1) N-1 SUM 8 7 6 5 4 3 2 1 k N
1 1 1 1
4 4 3 1 2
11 11 7 3 1 3
30 30 19 7 3 1 4
85 85 55 19 7 3 1 5
248 248 163 55 19 7 3 1 6
735 735 487 163 55 19 7 3 1 7
2194 2194 1459 487 163 55 19 7 3 1 8
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The 67 Algorithm Duration Ratio
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B3. The Free MToH The 62 Algorithm
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The 62 Algorithm Number of Moves
SUM 8 7 6 5 4 3 2 1 k N
1 1 1
4 3 1 2
11 7 3 1 3
30 19 7 3 1 4
83 53 19 7 3 1 5
236 153 53 19 7 3 1 6
691 455 153 53 19 7 3 1 7
2050 1359 455 153 53 19 7 3 1 8
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The 62 Algorithm Duration Ratio
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SF 67 62 Duration Ratio
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SF 67 62 Duration-Ratio Curves
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The double-pan balance Puzzle
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Effective (minimum of) weights for a balance
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Minimum of weights - continue
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9
3
1
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Minimum of weights - continue
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Elegance of the 67 Algorithm
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The 67 Algorithm find a simple rule
SUM 8 7 6 5 4 3 2 1 k N
1 1 1
4 3 1 2
11 7 3 1 3
30 19 7 3 1 4
85 55 19 7 3 1 5
248 163 55 19 7 3 1 6
735 487 163 55 19 7 3 1 7
2194 1459 487 163 55 19 7 3 1 8
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What about the total number of moves?
The Free 67 Magnetic Tower of Hanoi Total
number of moves
SUM N
1 1
4 2
11 3
30 4
85 5
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Recursive Relations
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Recursive Relations - 1
The 100 Algorithm
The 67 Algorithm
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Recursive Relations - 2
The SF Algorithm
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Recursive Relations - 3
The 62 Algorithm
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Recursive Relations - 4
All without exception
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Color Crossings
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MToH Color Crossings - 1
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MToH Internet Movie
A "movie" showing the "62" Algorithm solving a
height five MToH in (only) 83 moveshttp//www.nu
merit.com/maghanoi/
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MToH Internet Movie
Shown in the movie solution of the height 5
MToH puzzle by (only) 83 moves
Click link to play a YouTube video 6.
http//www.youtube.com/watch?vsysN4-6zXNo
It is Freedom that makes the MToH so colorful.
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MToH Color Crossings - 2
The 100 Algorithm NO color crossings
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MToH Color Crossings - 3
The 62 Algorithm EIGHT color crossings
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Next
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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?

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References
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Gathering 4 Gardner 9 Atlanta, GA (March 10)
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Gathering 4 Gardner 9 Atlanta, GA (March 10)
Game inventor Martin Gardner
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Gathering 4 Gardner 9 mini-MToH
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G4G9 - Handouts
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References
1 "The Magnetic Tower of Hanoi", Uri
Levy, Journal of Recreational Mathematics
353, to be published (May 2010) 2 Paper
download http//arxiv.org/abs/1003.0225 3 "M
ovie (and paper download, different Abstract)
http//www.numerit.com/maghanoi 4 Contact
uri_at_vicsor.com
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Cornell University Library
http//arxiv.org/abs/1003.0225
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Realization
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The Magnetic Tower of Hanoi Realization
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The Colored Magnetic Tower of Hanoi
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Free or Classical MToH
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Oops!
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Illegal Move!
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One-Two-Three GO!
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The End