The Magnetic Tower of Hanoi
1
The Classical Tower of Hanoi
2
Classical ToH – video-clip 1 (0:53)
Click link to play a YouTube video
1. http://www.youtube.com/watch?v=EHtk7kZqoVY
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
[
]
(Pos. – 1)
Number = SUM npos.*Base
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
1
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 [ToH]
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 ToH – video-clip 2 (1:14)
Click link to play a YouTube video
2. http://www.youtube.com/watch?v=3eGBhSSxffM
11
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)
12
ToH – Recursive Relations
P ( k  1)  2  P ( k )
S ( N  1)  2  S ( N )  1
13
ToH – Number of Moves
P (k )  2
N
S (N ) 
2
k 1
k 1
 2
N
1
k 1
k
N
1
2
3
4
5
6
7
1
1
2
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
1
2
4
8
16
32
64
8
128
SUM
2N - 1
1
1
3
3
7
7
15
15
31
31
63
63
127
127
255
255
14
Classical ToH – 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 2N - 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
MToH – video-clip 3 (1:29)
Click link to play a YouTube video
3. http://www.youtube.com/watch?v=nUoHHeaJ4eI
22
B. The Magnetic Tower of Hanoi [MToH]
23
MToH – when we where Young and Brave
Yaron (10) and a home-made Magnetic Tower of Hanoi
Rehovot, Israel - Autumn 1984.
24
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)
25
MToH – Solving the N=2 Puzzle
2
1
4
3
26
Colored MToH – video-clip 4 (1:24)
Click link to play a YouTube video
4. http://www.youtube.com/watch?v=D_xfuCOh1S0
27
B1. The Colored MToH
S
D
I
D
I
S
I
D
28
The Colored MToH – Number of Moves
P100 ( k )  3
N
S 100 ( N ) 
3
k 1
k 1

k 1
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
7
1
3
9
27
81
243
729
8
1
3
9
27
81
243
729
3
N
1
3 1
1
8
2187
SUM
(3N - 1)/2
1
1
4
4
13
13
40
40
121
121
364
364
1093
1093
3280
3280
29
Colored MToH – 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
(3N–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
MToH – The Three Versions
B1. The Colored MToH
B2. The Semi-Free MToH
B3. The Free MToH
32
B2. The Semi-Free MToH
SS
ID
DI
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
33
The Semi-Free MToH – Number of Moves
k - odd
PSF ( k )  2  3
k 2

3
k 1
k - even
3
2

3 1
2
3
k 2
3
1
3 1
2
 1 PSF ( k )  2  3
N - odd
S SF ( N )  ( 3
k
N
N 1
1
 N  1) 
2
k 2

3
k 1
3
1
3 1
2

3
k 2
3
2
3 1
2
N - even
3
N 1
3
3 1
2
3
0

N 1
2
4
S SF ( N )  ( 3
5
N 1
6
 N  1) 
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
1
3
7
21
61
183
547
3
N 1
3
3 1
2
8
1

N 2
2
SUM
1
4
11
32
93
276
823
1641
2464
34
The Semi-Free MToH – Duration Ratio
N 
S SF ( N ) S 100 ( N )    3 4
35
The Free MToH
•
The “67” Algorithm
•
The “62” Algorithm
36
The Free MToH – video-clip 5 (2:43)
Click link to play a YouTube video
5. http://www.youtube.com/watch?v=bZtx5gexddI
37
MToH FREEDOM
“It is (this) FREEDOM
that makes the
Magnetic Tower of Hanoi Puzzle
so
COLORFUL”
38
B3. The Free MToH – The “67” Algorithm
P67 (1)  1
P67 ( k )  2  P100 ( k  1)  1
k  2
S 67 ( N )  S 67 ( N  1)  4  S 100 ( N  2 )  3
39
The “67” Algorithm – Number of Moves
P67 (1)  1
P67 ( k )  2  3
 2  3
N
S 67 ( N )  1 
k 2
k 2

1  3
1 k  2
N 1
 N 1
k 2
k
N
1
2
3
4
5
6
7
1
1
2
1
3
3
1
3
7
4
1
3
7
19
5
1
3
7
19
55
6
1
3
7
19
55
163
7
1
3
7
19
55
163
487
8
1
3
7
19
55
163
487
8
1459
SUM
3(N-1) + N-1
1
1
4
4
11
11
30
30
85
85
248
248
735
735
2194
2194
40
The “67” Algorithm – Duration Ratio
N 
S 67 ( N ) S 100 ( N )    2 3
41
B3. The Free MToH – The “62” Algorithm
P62 ( k )  P67 ( k )
k 3
P62 ( k )  2  P100 ( k  2 )  2  P100 ( k  3 ) 
k 3
 P67 ( k  1)  P67 ( k  2 )  PSF ( k  3)
S 62 ( N )  S 67 ( N )
N 3
S 62 ( N )  2  S 100 ( N  2 )  2  S 100 ( N  3 ) 
3 1
2
 S 67 ( N  1)  S 67 ( N  2 )  S SF ( N  3 ) 
N 3
3 1
1
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
5
1
3
7
19
53
6
1
3
7
19
53
153
7
1
3
7
19
53
153
455
8
1
3
7
19
53
153
455
8
SUM
1
4
11
30
83
236
691
1359
2050
43
The “62” Algorithm – Duration Ratio
N 
S 62 ( N ) S 100 ( N )    67 108
44
“SF” ; “67” ;“62” – Duration Ratio
S SF ( N )
N 
  
3
N 1
S 100 ( N )
3
N 1
3
N
2 1  ( N  1) / 3
 
N
S 100 ( N ) 3
11/ 3
S 67 ( N )
S 62 ( N )
S 100 ( N )
N 
  
2 3
N 2
( 3  1)
2

2
3
4
( N 1 )
n 


 2 3
2
3
N 3
3
3
N
N 4
2

 3
N 4
( 3  1)
2

67
108
45
“SF” ; “67” ; “62” – Duration-Ratio Curves
"SF", "67", "62" efficiency curves
1
"SF"
"SF" limit
"67"
"67" limit
"62"
"62" limit
0.95
S-XX(N)/S-100(N)
0.9
0.85
0.8
3/4
0.75
2/3
0.7
0.65
0.6
67/108
0.55
0.5
1
2
3
4
5
6
7
8
9
10
11
12
Number of disks in the stack (N)
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
81
3
1 through 121
27
50
Elegance of
the
“67 Algorithm”
51
The “67” Algorithm – find a simple rule
k
N
1
1
2
3
1
2
3
4
5
6
7
8
SUM
1
4
11
1
3
1
3
7
4
5
6
1
3
7
19
1
3
7
19
55
1
3
7
19
55
163
7
8
1
3
7
19
55
163
487
1
3
7
19
55
163
487
P67 ( k )  2  3
30
85
248
k 2
1459
735
2194
1
52
What about the total number of moves?
The “Free 67” Magnetic Tower of Hanoi
Total number of moves
N
1
2
SUM
S 67 ( N )  3
N 1
 N 1
1
S 67 (1)  3
4
S 67 ( 2 )  3  1  4
0
 0 1
1
11
S 67 ( 3 )  3
2
 2  11
4
30
S 67 ( 4 )  3
3
 3  30
5
85
S 67 ( 5 )  3
4
 4  85
3
53
Recursive
Relations
54
Recursive Relations - 1
The “100” Algorithm
P100 ( k  1)  3  P100 ( k )
S 100 ( N  1)  3  S 100 ( N )  1
The “67” Algorithm
P67 ( k  1)  3  P67 ( k )  2
k  2
S 67 ( N  1)  3  S 67 ( N )  2  N  3
55
Recursive Relations - 2
The “SF” Algorithm
PSF ( k  1)  3  PSF ( k )
k - odd
PSF ( k  1)  3  PSF ( k )  2
k - even
S SF ( N  1)  3  S SF ( N )  N  1
N - odd
S SF ( N  1)  3  S SF ( N )  N  2
N - even
56
Recursive Relations - 3
The “62” Algorithm
P62 ( k  1)  3  P62 ( k )  6
P62 ( k  1)  3  P62 ( k )  4
k  4
k  4
k - odd
k - even
S 62 ( N  1)  3  P62 ( N )  5  ( N  3 )  3 N  3
S 62 ( N  1)  3  P62 ( N )  5  ( N  3 )  2 N  3
N - odd
N - even
57
Recursive Relations - 4
All without exception:
  next   3   ( current )  ...
58
Color Crossings
59
MToH – Color Crossings - 1
Color of a given post = Red → Neutral →
{
Red
Blue
60
MToH – Internet Movie
A "movie" showing the "62" Algorithm solving a height five MToH in (only) 83 moves:
http://www.numerit.com/maghanoi/
61
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?v=sysN4-6zXNo
It is Freedom that makes the MToH so colorful.
62
MToH – Color Crossings - 2
P o st c o lo r [1 = R ; 0 = N ; -1 = B ]
The "100" Algorithm ; N = 5 [121 moves]
1
Source
Intermediate
Destination
0
-1
0
10
20
30
40
50
60
70
80
90
100
110
120
Move-number
The “100” Algorithm – NO color crossings
63
MToH – Color Crossings - 3
P o st c o lo r [1 = R ; 0 = N ; -1 = B ]
The "62" Algoritm ; N = 5 [83 moves]
1
0
Source
Intermediate
Destination
-1
0
10
20
30
40
50
60
70
80
Move-number
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-MToH
70
G4G9 - 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” MToH
77
Oops!
78
Illegal Move!
79
One-Two-
Three – GO!
80
The End
81