Transcript The Tower Of Hanoi - 2July

The Tower Of Hanoi
Edouard Lucas - 1883
Once upon a time!
• The Tower of Hanoi (sometimes referred to
as the Tower of Brahma or the End of the
World Puzzle) was invented by the French
mathematician, Edouard Lucas, in 1883. He
was inspired by a legend that tells of a
Hindu temple where the pyramid puzzle
might have been used for the mental
discipline of young priests.
What does legend say??
• Legend says that at the beginning of time the priests in
the temple were given a stack of 64 gold disks, each
one a little smaller than the one beneath it. Their
assignment was to transfer the 64 disks from one of
three poles to another, with one important proviso a
large disk could never be placed on top of a smaller
one. The priests worked very efficiently, day and
night. When they finished their work, the myth said,
the temple would crumble into dust and the world
would vanish. How many moves ( & how long) would
the priests need to take to complete the task??
Start here - Instructions
1. Transfer all the disks from one pole to another pole.
2. You may move only ONE disk at a time.
3. A large disk may not rest on top of a smaller one at any
time.
1
2
3
A
B
C
Try this one!
1
2
3
4
A
B
C
Shortest number of moves??
And this one
1
2
3
4
5
A
B
Shortest number of moves??
C
Now try this one!
1
2
3
4
5
6
A
B
Shortest number of moves??
C
Where's the maths in this game?
• From the moves necessary to transfer one, two, and three
disks, we can find a recursive pattern - a pattern that uses
information from one step to find the next step.
• Unfortunately, if we want to know how many moves it will
take to transfer 64 disks from post A to post C, we will first
have to find the moves it takes to transfer 63 disks, 62
disks, and so on. Therefore the recursive pattern will not be
much help in finding the number of moves or the time it
would take to transfer all the disks.
• However, the recursive pattern can help us generate more
numbers to find an explicit (non-recursive) pattern. Here's
how to find the number of moves needed to transfer larger
numbers of disks from post A to post C, when M = the
number of moves needed to transfer n-1 disks from post A
to post C:
• for 1 disk it takes 1 move to transfer 1 disk from post A to
post C;
•
•
•
•
•
for 2 disks, it will take 3 moves: 2M + 1 = 2(1) + 1 = 3
for 3 disks, it will take 7 moves: 2M + 1 = 2(3) + 1 = 7
for 4 disks, it will take 15 moves: 2M + 1 = 2(7) + 1 = 15
for 5 disks, it will take 31 moves: 2M + 1 = 2(15) + 1 = 31
for 6 disks... ?
Explicit Pattern
• Number of Disks
1
2
3
4
5
6
Number of Moves
1
3
7
15
31
63
Powers of two help reveal the pattern:
• Number of Disks (n)
1
2
3
4
5
6
Number of Moves
21 - 1 = 2 - 1 = 1
22 - 1 = 4 - 1 = 3
23 - 1 = 8 - 1 = 7
24 - 1 = 16 - 1 = 15
25 - 1 = 32 - 1 = 31
26 - 1 = 64 - 1 = 63
Fascinating fact
So the formula for finding the number of steps it takes to
transfer n disks from post A to post C is:
2 n- 1
• The number of separate transfers of single disks the priests
must make to transfer the tower is 2 to the 64th minus 1, or
18,446,744,073,709,551,615 moves! If the priests worked
day and night, making one move every second it would
take slightly more than 580 billion years to accomplish the
job! - far, far longer than some scientists estimate the solar
system will last.
The Tower Of Hanoi