Borders Coursework

Trying for a formula - white squares.

In each case I have observed that if you multiply the pattern number by 4 it gives you the amount of white squares.
E.g.
Pattern
1 x 4 = 4 white squares
2 x 4 = 8 white squares
3 x 4 = 12 white squares
This goes on & by using this method you can find the amount of white squares as long as you have the pattern number.

I have also noticed that the new addition under the new orders table gives the amount of white squares. This is also true for the 1st difference.

Formula for white squares

Throughout my investigation I will use the following symbols for the formula's, these will not change.
N = pattern number
D = dark squares
W = white squares

I think the formula is simply 4 x the pattern number or 4N. The following is a check to see if my formula is correct.

4N = number of white squares
N = 10
4 x 10 = 40
This is the correct amount of white squares; I know this because of the table of new orders gives me this answer.

Finding a formula for the dark squares
  I am now searching for a formula for the dark squares.
I will analyse the information of the tables of new orders & differences find a formula for the amount of dark squares. Once again I will use the same letters for the pattern number etc.
 
I have found that the number of dark squares equals the total for the previous squares. To see this you can look on the bottom of page 1.
I have decided to draw a new table similar to that of the table of differences on page 2 to help me find the progression of dark squares.
Pattern Total squares Dark squares White squares
1 5 1 4
2 13 5 8
3 25 13 12
4 41 25 16
5 61 41 20
6 85 61 24
7 113 85 28
8 145 113 32
9 181 145 36
10 221 181 40
There are only three important columns in this table those are the first three. The column of white squares is only there because it shows how I got the amount of total squares.

The differences between the dark & total number of squares once again go up in 4.
I think to find a formula for the dark squares you can find a possible formula to find the amount of total number of squares & then minus the formula for the white squares.
E.g.
Total number of squares - 4N = number of dark squares
Adding to my list of algebraic letters I will add the letter T.
N = pattern number
D = dark squares
W = white squares
T = total number of squares

I am now going to complete a table to show the development of the pattern number (N) against the total number of squares (T).
N T
1 5 + 4
2 13 + 11
3 25 + 22
4 41 +37
5 61 +56
6 85 +79
7 113 +
8 145 +
9 181 +
10 221 +

Order of squares going down1 + 3 + 1 = 51 + 3 + 5 + 3 + 1 = 13New additions. In each case 2 has been added.1 + 3 +5 7 + 5 + 3 + 1 = 25Like the previous pattern there are Additions with 2 being added on.For the next pattern I predict that the total number of squares will be 41, using the following pattern:1 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41I am now going to check to test my prediction.Number of squares = 25 + 16 = 41My prediction was correct. As well as finding a correct method of finding the next pattern I noticed that to find the number of dark squares on the next pattern you use the total number for the previous pattern.Pattern Dark squares White squares
1 1 4 1 + 4 = 5
2 5 8 5 + 8 = 13
3 13 12 13 + 12 = 25
4 25 16 25 + 16 = 41
WHY?This happens because you are simply adding on to this. This could be a useful fact in searching for a formula.I am now going to investigate any differences between the totals.First of all I will need to find some more totals, as the amount I have will not be conclusive. To find these without drawing any more diagrams I will use my knowledge of the structure (e.g. 1 + 3 + 1).New ordersPattern Previous total New additions New totals
5 41 11 + 9 = 20 61 (20 + 41 = 61)
6 61 13 + 11 =24 85 (61 + 24 =85)
7 85 15 + 13 = 28 113
8 113 17 + 15 =32 145
9 145 19 + 17 = 36 181
10 181 21 + 19 =40 221
                                                                                     (These are white squares)DifferencesTotal 5 13 25 41 61 85 113 145 181 221
1st difference 8 12 16 20 24 28 32 36 40
2nd difference 4 4 4 4 4 4 4 4
 This shows a main difference of 4. I think this will influence the formula. I think this will mainly be in the form of a multiple of 4.The first formula I will try to find is the formula for the surrounding white squares. Trying for a formula – white squares.In each case I have observed that if you multiply the pattern number by 4 it gives you the amount of white squares.E.g.Pattern1 x 4 = 4 white squares2 x 4 = 8 white squares3 x 4 = 12 white squaresThis goes on & by using this method you can find the amount of white squares as long as you have the pattern number.I have also noticed that the new addition under the new orders table gives the amount of white squares. This is also true for the 1st difference.Formula for white squaresThroughout my investigation I will use the following symbols for the formula´s, these will not change.N = pattern numberD = dark squaresW = white squaresI think the formula is simply 4 x the pattern number or 4N. The following is a check to see if my formula is correct.4N = number of white squaresN = 104 x 10 = 40This is the correct amount of white squares; I know this because of the table of new orders gives me this answer.Finding a formula for the dark squares I am now searching for a formula for the dark squares.I will analyse the information of the tables of new orders & differences find a formula for the amount of dark squares. Once again I will use the same letters for the pattern number etc. I have found that the number of dark squares equals the total for the previous squares. To see this you can look on the bottom of page 1.I have decided to draw a new table similar to that of the table of differences on page 2 to help me find the progression of dark squares.Pattern Total squares Dark squares White squares
1 5 1 4
2 13 5 8
3 25 13 12
4 41 25 16
5 61 41 20
6 85 61 24
7 113 85 28
8 145 113 32
9 181 145 36
10 221 181 40
There are only three important columns in this table those are the first three. The column of white squares is only there because it shows how I got the amount of total squares.The differences between the dark & total number of squares once again go up in 4.I think to find a formula for the dark squares you can find a possible formula to find the amount of total number of squares & then minus the formula for the white squares.E.g.Total number of squares – 4N = number of dark squaresAdding to my list of algebraic letters I will add the letter T.N = pattern numberD = dark squares W = white squaresT = total number of squaresI am now going to complete a table to show the development of the pattern number (N) against the total number of squares (T).N T
1 5 + 4
2 13 + 11
3 25 + 22
4 41 +37
5 61 +56
6 85 +79
7 113 +
8 145 +
9 181 +
10 221 +|