6*4+6=30
6th Border
As you can see there are 30 squares numbered 6 indicating that my prediction is right. This shows that my rule is correct.
1 BY 5
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 12. In this case it isn’t so I will have to add 8 to get to 12. So my rule is 4B+8. Using this rule I predict that the number of numbered squares in the 6thborder is 32.
6*4+8=32
6th Border
As you can see there are 32 squares numbered 6 indicating that my prediction is right. This shows that my rule is correct.
2 BY2
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 8. In this case it isn’t so I will have to add 4 to get to 8. So my rule is 4B+4. Using this rule I predict that the number of numbered squares in the 6thborder is 28.
6*4+4=28
6th Border
As you can see there are 28 squares numbered 6 indicating that my prediction is right. This shows that my rule is correct.
2 BY 3
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 10. In this case it isn’t so I will have to add 6 to get to 10. So my rule is 4B+6. Using this rule I predict that the number of numbered squares in the 6thborder is 30.
6*4+6=30
6th border
As you can see there are 30 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
2 BY 4
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 12. In this case it isn’t so I will have to add 8 to get to 12. So my rule is 4B+8. Using this rule I predict that the number of numbered squares in the 6thborder is 32.
6*4+8=32
6th Border
As you can see there are 32 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
2 BY 5
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 14. In this case it isn’t so I will have to add 10 get to 14. So my rule is 4B+10.Using this rule I predict that the number of numbered squares in the 6thborder is 34.
6*4+10=34
6th Border
As you can see there are 34 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
3 BY 3
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 12. In this case it isn’t so I will have to add 8 get to 12. So my rule is 4B+8.Using this rule I predict that the number of numbered squares in the 6thborder is 32.
6*4+8=32
6th Border
As you can see there are 32 squares numbered 6 indicating that my prediction is right. This shows that my rule is correct.
3 BY4
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 14. In this case it isn’t so I will have to add 10 get to 10. So my rule is 4B+10.Using this rule I predict that the number of numbered squares in the 6thborder is 34.
6*4+10=34
6th Border
As you can see there are 34 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
3 BY 5
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 16. In this case it isn’t so I will have to add 12 get to 16. So my rule is 4B+12.Using this rule I predict that the number of numbered squares in the 6thborder is 36.
6*4+12=36
6th Border
As you can see there are 36 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
4 BY 4
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 16. In this case it isn’t so I will have to add 12 get to 16. So my rule is 4B+12.Using this rule I predict that the number of numbered squares in the 6thborder is 36.
6*4+12=36
6th Border
As you can see there are 36 squares numbered 6 signifying that my prediction is right. This shows that my rule is correct.
4 BY 5
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 18. In this case it isn’t so I will have to add 14 get to 16. So my rule is 4B+14.Using this rule I predict that the number of numbered squares in the 6thborder is 38.
6*4+14=38
6th Border
As you can see there are 38 squares numbered 6 showing that my prediction is right. This shows that my rule is correct.
5 BY 5
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 20. In this case it isn’t so I will have to add 16 get to 20. So my rule is 4B+16.Using this rule I predict that the number of numbered squares in the 6thborder is 40.
6*4+16=40
6th Border
As you can see there are 40 squares numbered 6 which shows that my prediction is right. This shows that my rule is correct.
5 BY 6
Using my table of results I can work out a rule finding the term-to-term rule. With the term-to-term rule I can predict the 6th border. As you can see, number of numbered squares goes up in 4. To work out the rule I will multiply 1 by 4. Then I will see if the answer is 22. In this case it isn’t so I will have to add 18 get to 22. So my rule is 4B+18.Using this rule I predict that the number of numbered squares in the 6thborder is 42.
6*4+18=42
6th Border
As you can see there are 42 squares numbered 6 which shows that my prediction is right. This shows that my rule is correct.
Overall Formulas
How all the overall formulas were found?
Y by 1, which is:
S=4b+2y-2
All the sequence formulas included 4b in them. As a result of this the general rule has 4b in it because it is the most recurrent. Following that I apply 1x y and the sequence increases by 2. Therefore the common rule must have +2y. At this time I have to find out the difference so that I know if I have to add or deduct from the formula. In this case I have to minus 2 to the general rule.
Y by 2, which is:
S=4b+2y
I become aware of the fact that the sequences has 4b, this is the most recurrent and therefore goes into the common rule and they increase by 2 so therefore it must be s=4b+2y. This is the general rule for 2xy because there is no need to minus or add to the formula and therefore stays the same.
Y by 3 which is:
S=4b+2y+2
As 4b is the most occurring, in fact it occurs in all the sequence formulas, it is automatically inputted in the universal rule. It also increases by 2 therefore +2y is applied in the universal. So far the general rule stands at 4b+2y it now needs the difference added or subtracted in this case I will need to add +2.
Y by 4 which is:
S=4b+2y+4
Once more I notice that 4b is the most recurrent and occurring so it is automatically in the general rule. After this I find out the dissimilarity between the sequences formulas and this is +2. Therefore I have to apply +2y. The formula is not accurate so therefore I need to find the difference needed to add or subtract and this was +4.
Y by 5 which is:
S=4b+2y+6
Looking back at all the other formulas I have analyzed and recognized that all the general rules are going up in two’s. I have looked at the previous formula which is 4by Y and the formula there is S=4b+2y+4 so I will have to add 2 to that in order for me to find out the general rule for Y by 5.
Finding the general rule for X by Y
General rule for x x y: s=4b+2y+2x-4
First of all I have noticed that 4b+2y is the most recurrent in the general rules so therefore is automatically transferred in the general rule for x by y. I have now found the border number (b) and the number of columns. Now I need to find the number of rows this is found out according to the variation, which indicates 2 therefore it, is +2x because it is linear. Finally I have to find the variation in the arithmetic’s and in this case I have to minus 4.
Testing the general rule for x x y
Testing is very important so that I know my general rule is correct. So I have devised a plan of random numbers that will be tested. They will each be from different borders, rows and columns.
b=3 x=4 y=5 the answer should be 26.
4(3) +2(4)+2(5)-4=26
b=2 x=5 y=3 the answer should be 20
4(2)+2(5)+2(3)-4=20
b=3 x=2 y=3 the answer should be 18
4(3) +2(2) +2(3)-4=18
As you can see from the testing from random numbers that is between 1 to 5 the general rule for x by y has clearly worked. Therefore s=4b+2y+2x-4 is the general rule to find out anything in the border patterns as long as there is a border number, row number and column.
Conclusion
In the time available to me, I believe that I have researched the links between the borders of each square to the full extent of my ability. I think the solution was found in a legitimate and a reasonable way. I found a formula which links the borders of each square. I then extended the squares and was able to construct my general formula, which will tell you the size of each border of any square constructed. I found that the amount of borders of a square must be taken into account in the formula. I also found that many of my predictions I made along the way turned out to be correct.
I am certain that the reason my general formula ends (4B+a number) and not just B is because when taking the number of borders of the square into account, you only want to know the ones that are touching each other. It can be extended into quadratic terms.
I would say that this investigation has been a success. I managed to find a link between borders. After that I went a stage further and developed a formula that would tell you the border of any square possible.
Unfortunately, my investigation was hindered by 1 thing – foremost was my lack of time to carry out the investigation as far as possible (i.e. researching 3D squares).
If I were to redo this investigation, I would make sure that I set aside enough time to do a proper job of it. I also would have very much liked to have been able to move on to looking at 3D squares, but doing this means wither physically making the square’s border out of cube or something else, which is vastly time-consuming. I could have done the square border with multi-colored cubed, which was my best option but my predicament was that I did not have any.
