1)
Moves: 1-C
My prediction was correct and I now can confirm it takes me 1 move to successfully move 1 disc from start to finish.
Investigation 3
Now I am going to try to successfully move 2 discs from start (A) to finish (B or C) in the least number of moves possible.
This is the position I am going to start my challenge form.
1) 2)
3)
Moves: 1-B
2-C
1-C
My challenge was successful and I completed it in 3 moves.
Investigation 4
Now in my fourth investigation I am going to attempt to move 3 discs from start (A) to finish (B or C) in the minimum amount of moves possible.
The position I am going start from is at the bottom of the last page.
1) 2)
3) 4)
5) 6)
7)
Moves: 1-C
2-B
1-B
3-C
1-A
2-C
1-C
My challenge was successful as I moved 4 discs from start (A) to finish (B) or (C.)
Investigation 5
Now in my fifth investigation I am going to attempt to move 5 discs from start (A) to finish (B or C.)
Moves: 1-B
2-C
1-C
3-B
1-A
2-B
1-B
4-C
1-C
2-A
1-A
3-C
1-B
2-C
1-C
5-C
1-A
2-C
1-C
3-A
1-B
2-A
1-A
4-C
1-C
2-B
1-B
3-C
1-A
2-C
1-C
My challenge was successful as I moved the 5 discs from start (A) to the finish (B or C.) I looked for a pattern and noticed the difference between the number of moves for the different number of discs is doubled.
Here’s the table of results so far: -
As we can see the difference between the number of moves 1 and 3 is 2 and then between 3 and 7 is 4. Here we can see the number has doubled. The difference between 7 and 15 is 8 (the number has again doubled) and the number of moves again doubles between 15 and 31, as it is 16. Now I can predict how many moves it will take for 6 discs to get from the start (A) to the finish (B or C).
Investigation 6
I will now investigate the sixth challenge and I will try to successfully move 6 discs from the start (A) to the finish (B or C.)
Prediction
I can now predict the number of moves that it will take me to move all 6 discs from start to finish.
Here is the pattern I am going to use: -
2 x 1 – 1 = 1
2 x 2 – 1 = 3
2 x 2 x 2 –1 = 7
2 x 2 x 2x 2 – 1 = 15
2 x 2 x 2 x 2 x 2 –1 = 31
2 x 2 x 2 x 2 x 2 x 2 – 1 = 63
Moves: 1-B
2-C
1-C
3-B
1-A
2-B
1-B
4-C
1-C
2-A
1-A
3-C
1-B
2-C
1-C
5-C
1-A
2-C
1-C
3-A
1-B
2-A
1-A
4-C
1-C
2-B
1-B
3-C
1-A
2-C
1-C
6-C
1-C
2-A
1-A
3-C
1-B
2-C
1-C
4-A
1-A
2-B
1-B
3-A
1-C
2-A
1-A
5-C
1-B
2-C
1-C
3-B
1-A
2-B
1-B
4-C
1-C
2-A
1-A
3-C
1-B
2-C
1-C
My prediction is correct and so it is possible to move all 6 discs from start to finish in a minimum of 63 moves, which also confirms that my pattern works. This enables me to work out the number of moves it will take for any number of discs. This now means I can now construct a formula, which I will do later on in my results.
Table Of Results
This table of results clearly shows us the results we found and also the table is quite easy to understand and read.
Looking for a pattern
The pattern I found was that the difference between the number of moves doubles each tome. For example between 1 and 3 the number is 2, then between 3 and 7 the number is 4 and between 7 and 15 is 8 and so on the pattern continues.
2 x 1 = 2
2 x 2 = 4
2 x 4 = 8
2 x 8 = 16
2 x 16 = 32
2 x 32 = 64
2 x 64 = 128
This pattern enables me to predict the next number as all I do is add the doubled number onto the last number of moves. For example for 6 discs the number of moves taken were 63 and so I can add on 64 to 63 and I get the number of moves needed to complete 7 discs.
Graph of results
This graph will be on the next page.
Finding a rule and checking it
I found 2 types of rules which work and are successful.
- Position to term rule:
This is a rule which enables me to predict the amount of moves it will take for any number of discs. The rule is 2n-1. This rule works by me multiplying the number by itself by different numbers and then subtracting 1 each time.
E.g.: -
2 –1 = 1
2 x 2 – 1 = 3
2 x 2 x 2 – 1 =7
2 x 2 x 2 x 2 – 1 = 15
2 x 2 x 2 x 2 x 2 – 1 = 31
2 x 2 x 2 x 2 x 2 x 2 – 1 = 63
2 x 2 x 2 x 2 x 2 x 2 x 2 – 1 = 127
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 – 1 = 255
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 –1 = 511
As you can see all these sums work and I could carry on using the formula.
-
Term to term rule:
This is a rule which enables me to find the amount of moves required for any number of discs. The rule is 2n+1. All I do is double the last term, For example: -
2 x 0 +1 = 1
2 x 1 +1 = 3
2 x 3 +1 = 7
2 x 7 +1 = 15
2 x 15 +1 = 31
2 x 31 +1 = 63
2 x 63 +1 = 127
2 x 127 +1 = 255
2 x 255 +1 = 511
2 x 511 +1 = 1023
As you can see the rule works well and so is very helpful in finding the next trem along. All you do is double the last term like double the 3 and +1 you get 7 and then you do the same again.
Conclusion
What I discovered is that there are simple ways of solving these investigations. These patterns and rules were really helpful in the end for my work as when I was trying to crack one of the puzzles I knew how many moves I had to do it in which help. I knew this due to the patterns I found and the rules. The 2 rules I found were the 2n-1 rule, which is a position to term rule, and the other rule, which is the term-to-term rule, is 2n+1. Overall I found that finding the rules and patterns came quite easily in the end as the patterns built up as I done more work.
