Borders Maths Investigation - page 2
Keywords: Borders Maths Investigation
By Grant867 on 07/12/2008
Level: GCSE Key Stage 4 (Years 10-11)
Page Number: 2 of 3 pages: 1 2 3
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.
Pattern
'font-size:12.0pt; '>1 x 4 = 4 white squares
'font-size:12.0pt; '>2 x 4 = 8 white squares
'font-size:12.0pt; '>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
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