The Tower of Hanoi is a famous mathematical puzzle.
Rules: you have to move the entire stack of discs, but you can only move one disc at a time, and you cannot stack a disc onto a smaller disc. It's trickier than it looks!
Click the following link to try the interactive puzzle:
http://www.mazeworks.com/hanoi/index.htm
Start with 3 discs, and see if you can solve the puzzle in the minimum number of moves. Then try 4, then 5, and so forth!
Comment below to answer any of the following questions:
* Was the puzzle difficult to complete? Explain.
* What was the greatest number of discs you were able to move? Were you able to solve the puzzle in the minimum number of moves?
* Did you develop a strategy? If so, try to describe it.
** Advanced: Is there a relationship between the number of discs and the number of minimum moves?
Please remember to sign your comment with your first name and last initial - no last names!
Lillian B. Period 4
ReplyDeleteThe puzzle got more difficult to complete with the more blocks that were added.
I was able to do 5 blacks, in the minimum number of moves.
My strategy was to put the blocks that were only one size smaller on top of the other blocks. When the sizes were more than just one block different, it was harder to do. However, the smallest block had to be put on top of the larger and sometimes the largest block occasionally, especially toward the end of the puzzle.
The minimum number of moves required has to get larger as the number of blocks grows because each block has to be moved numerous times in order to succeed. I found that from 3-5, the minimum number of moves was very close to the number of blocks squared.
This is an example of a great post, Lillian! You answered several questions, provided good details, and demonstrated insight. Fantastic!
ReplyDeleteMiss S.
Abby R. Period 4
ReplyDeleteI was also able to do up to 5 blocks. However, I was never able to complete a puzzle in the minimum number of moves required. I thought the puzzle was very challenging.
At first, I didn't have a stradegy which is probably why I didn't do the minimum number of moves, but when I got to playing with 5 blocks I tried to get the biggest block all the way over to the furthest right ring. Usually after this I was only a few moves away from solving the puzzle.
Honestly, I did not see a correlation between the number of rings and the minimum amount of required moves until Lillie posted it above.
The puzzle is pretty challenging until you "get it". Abby, you're totally on the right track - the first thing I do when I work on The Tower of Hanoi is think about how I'm going to get the largest piece over to the right, and then I work backwards from there.
ReplyDeleteGreat post! I'm happy to see that you not only answered the questions, but also referred to another student's comment - this is what the blog is all about! :)
Griffin Period 4
ReplyDeleteI got to six blocks and it was very difficult to complete in the minimum number of moves. As the number of blocks increased, the number of moves increase and therefor so did the risks. It was a lot more upsetting to fail at getting the minimum of 7 blocks because repeating it would mean having to move the blocks 127 times(as compared to 7 times for the 3 blocks) without making a mistake to succeed. I can't even imagine getting to the 12 blocks and having to move them 4095 times!
My strategy was similar to Abby's and Miss S's. I would try to get the largest block to the right most peg first and the rest would be in their own stack on the middle peg. Then I would treat the middle stack as its own tower and try to get the bottom most block to the right. I would repeat this until I got all the blocks on the right most peg. I think the way to get the least amount of moves with more blocks is to plan your moves carefully because one wrong move could force you to restart. I found myself planing each tower out before the first move. I haven't devoted too much time to this but I found that the number of moves is increased by factor of 8 times 2. For example, in the set of three block the are 7 moves, in the example with 4 block there are 15 moves. 15-7= 8. Then if you multipy 8 by 2 and add that to 15 you get 31 which is the number of moves for 5 blocks.If you miltipy 16 by 2 and add it to 31 you get the number of moves for the 6th block set, 63. And so on a so forth...
Great work, Griffin!
ReplyDeleteI use the same process that you describe - focusing on getting the largest block to the right, then the next largest, and so forth. You describe your method in a clear and succinct manner. Good references to other comments!
The pattern you noticed definitely applies when you are finding the minimum number of moves. You can also say that when another block is added, you double the previous minimum and add one! <--- You may remember from Algebra II Honors that this is an example of a recursive formula.
Here is another question, for Griffin or anyone else: Can you come up with an explicit formula to find the number of minimum moves? Use "n" to represent the number of blocks. Hint: look at Griffin’s pattern. He’s VERY close!
Miss S.
Ali P. Period 4
ReplyDeleteBefore I quit, I had gotten to 8 blocks. As I got to the next level, they kept increasing in difficulty. I solved the puzzle for the minimum number of moves with 3, 4, and 5 blocks. However, it was too difficult to do that with 6 and 7 blocks. I did develop a strategy right from the beginning. I tried to move the biggest block to the right one pole at a time. It worked until I got to 6 blocks. Then, I developed a new strategy. I did numerous moves and then tried to get the biggest block all the way to the right pole. Then, I knew that I was only a matter of a few moves away from solving the puzzle. I did discover a relationship between the number of minimum moves and the number of blocks. I'll try to explain it a little simpler than Griffin. You multiply the increased factor of 8 by 2, then add it to the number of minimum moves the previous amount of blocks had. For example, in 3 blocks, there are 7 moves. In 4 blocks, there are 15 ((4*2) +7). In 5 blocks, there are 31 moves ((8*2) +15). In 6 blocks, there are 63 moves ((16*2) +31). In 7 blocks, there are 127 moves ((32*2) +63). In 8 blocks, there are 255 moves ((64*2) +127), and so on...