Tower of Hanoi

1
Tower of Hanoi

• Goal transfer all the rings on the left-hand
  post to the right-hand post in the fewest
  possible moves.
• Rules
• -you can only move one ring at a time
• -you cannot put a larger ring on top of a smaller
  one
• -moving a ring from one post to another is
  considered to be a move

2
Recursive Formula2tn-11

• Sometimes you are asked to find the least number
  of moves it takes a certain number of rings to be
  transferred from one post to another.
• To make it easier to find the least number of
  moves we suggest using the recursive formula.
• A recursive formula is a formula based on the
  previous answer or number of moves.

3
Recursive pattern

• Number of disks 1, number of moves 1
• Number of disks 2, number of moves3
• Example with one ring

4
Example with two rings
5
Congratulations!

• Based on what you just did you now know that it
  takes exactly one move to move one ring and that
  it takes three moves to move two rings.

6
Recursive pattern

• Now try to plug 3 in the formula to find the
  number of moves it takes to move three rings.
• 2(3)17 moves
• The formula tells us that it will take us at
  least 7 moves to transfer three rings to the
  other end (or post). Lets test it!

7
Example with three rings
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Explanation of the formula

• Now that you know that the formula works you
  might want to know why it works.
• The formula is 2tn-11
• It means that you time the previous number by two
  then add one to it.

9
Explanation of the formula

• It takes three moves to move two rings to the
  middle post. (or post B)

10
Explanation of the formula

• What we just did in the previous slide represents
  tn-1.
• tn-1 means the previous number of moves which is
  three in this case.
• It took us three moves to move the first two
  rings to the middle.

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Explanation of the formula

• The next thing you do when you have the two first
  rings in the middle is moving the last ring to
  opposite end.
• That is where you get 1 in the formula, by
  moving the extra ring to the other end.

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Explanation of the formula

• The last thing to do after moving that ring is
  to move the remaining two to the opposite post.

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Explanation of the formula

• This is where the 2tn-1 comes from. It takes
  three moves to move the two first rings to post B
  and three moves again to move them to post C. So
  since you do the exact same thing twice, you can
  multiply that thing by 2.

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Conclusion

• The recursive formula helps you to find the
  least number of moves and it is based on the
  previous moves.
• The formula works by plugging in your previous
  number of moves.

15
This presentation has been done

• By Onnie K.
• Michael O.