Hypothesis
I predict that there will be a difference between each cross-section shape.
Therefore there will be a quadratic sequence which can tell me how many squares would be needed in any given shape.
Page2
Prediction
1 + 3 + 1 = 5
1 + 3 + 5 +3 + 1 = 13
1 + 3 + 5 + 7 + 5 + 3 + 1 = 25
1 + 3 + 5 + 7 + 9 + 7 + 5 + 3 + 1 = 41
If you notice in my prediction there is a pattern of 2n-1 going to a certain point and then back again.
My 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.
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 orders
Page 3
Differences
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.
Finding a formula for the number of squares
2n2+2n+1
e.g.
n=2
2 x 22 + 2 x 2 + 1=13
Page 4
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
4 x 4 = 16 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.
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.
Page 5
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.
I have found that the number of dark squares equals the total for the previous squares.
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.
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.
Page6
2n2–2n + 1
e.g.
n=2
2 x 22 – 2 x 2 + 1= 5
Conclusion
In conclusion I found made a hypothesis to predict what I though of the differences in the cross-section shapes, this helped me to achieve the formula for the squares changes, the white squares changes and the dark squares changes, I also used tables and algebra sequences to help me find the formulas, they turned out to be very useful in helping me with my investigation.
