A dark cross-shape has been surrounded by white squares to create a bigger cross-shape. The next cross-shape is always made by surrounding the previous cross-shape with small squares. Investigate to see how many squares
Assignment
A dark cross-shape has been surrounded by white squares to create a bigger cross-shape. The next cross-shape is always made by surrounding the previous cross-shape with small squares. Investigate to see how many squares would be needed to make any cross-shape built up in this way.
Introduction
For my mathematics coursework I have been asked to do a two part investigation. Part one requires me to investigate a formula to see how many squares would be needed to make any two-dimensional cross-shape. The second part requires me to extend the investigation into three dimensions. Each time the cross shape increases in size, border squares are increased. These border squares cover the outside of the original cross shape thus creating a new, larger one. An example is shown below.
Equipment
In order to investigate the assignment, I will make use of Lego bricks to use them to build my shapes so that I can gather all of the information for finding the formula, which is a required part of this coursework. This will allow me to model the problem and gather preliminary data.
Two Dimensional Investigation
Investigating the 2D formula
I am going to start by building a 1x1 cross-shape and adding on the borders. The complete shape will be known as the cross-shape (this includes the border squares). The diagram for this cross-shape and the others that I investigated are shown below. The original shape is shown shaded, with the border squares clear.
A 1x1 cross-shape
(Shape 1)
A 3x3 cross-shape
(Shape 2)
A 5x5 Cross-shape
(Shape 3)
The order of squares going from left to right are:
Shape 2: 1 + 3 + 1 = 5 (see diagram below for example)
Shape 3: 1 + 3 + 5 + 3 + 1 = 13
In both cases, 2 has been added to the middle number (as shown below):
+ 3 +5 + 7 + 5 + 3 + 1 = 25
All of the above have so ...
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A 1x1 cross-shape
(Shape 1)
A 3x3 cross-shape
(Shape 2)
A 5x5 Cross-shape
(Shape 3)
The order of squares going from left to right are:
Shape 2: 1 + 3 + 1 = 5 (see diagram below for example)
Shape 3: 1 + 3 + 5 + 3 + 1 = 13
In both cases, 2 has been added to the middle number (as shown below):
+ 3 +5 + 7 + 5 + 3 + 1 = 25
All of the above have so far been correct. 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 = 41
A 7x7 Cross-shape
(Shape 4)
My prediction was correct. As well as discovering a correct method of finding the next pattern I noticed that to find the number of darker squares on the next pattern you use the total number for the previous pattern.
A 9x9 Cross-shape
(Shape 5)
Analysing Results
I think that investigating five different cross-shapes will give me enough information to find a formula. The data I have collected is shown in the table below.
Shape Sizes
Border Squares
Darker Squares
Total Squares
Single Sq.
0
x 1
4
5
3 x 3
8
5
3
5 x 5
2
3
25
7 x 7
6
25
41
9 x 9
20
41
61
From these results I can see a direct relationship with the amount of border squares. They increment by 4 every time the shape size increases. I think that this relationship will be important in finding the formula to discover how many border squares are required to surround the darker cross-shape thus making a bigger cross shape. I think this will mainly be in the form of a multiple of 4.
Another relationship with the results is that the original cross-shapes total squares is the same number as the total number of squares on the previous shape. For example, the total squares for a 5 x 5 cross-shape squares (including border squares) is 25 and the original Cross-shape squares (not including the border) for a 7 x 7 Cross-shape is also 25. Also the total amount of squares is always odd. I have also found that the number of dark squares equals the total for the previous cross-shapes squares. To see this you can look at the table below or the cross shapes near the beginning of this assignment.
Darker Squares
Total Squares
5
5
3
3
25
25
41
41
61
Developing a Formula
Using the formula a n 2 + b n + c (given) and the difference method below. I am able to show a pattern emerging.
Position in sequence: 0 1 2 3 4 5 6
No. Of Squares ( c ): 1 1 5 13 25 41 61 . . . . . (I)
First Differences ( a + b ): 0 4 8 12 16 20 . . . . . . (ii)
Second Differences (2a): 4 4 4 4 4 . . . . . . . . (iii)
The bottom row of differences indicates a constant number, which shows there to be a pattern. If the fourth row had not indicated a constant number pattern (i.e. 2,2,2,2 or 6,6,6,6) then I would keep increasing the rows until I found one. If I was struggling to find a constant number, I would gather more information. This is what happened with my 3D investigation.
If n is the position in the sequence, 2a = 4 (iii) (therefore) a = 4 / 2 a = 2.
a + b = 0 (ii) and a = 2
We have 2 + b = 0 b = -2 and c = 1 (I).
Substituting the above into the formula a n 2 + b n + c, we have:
a n 2 + b n + c
2 n 2 - (2n) + 1
2 n 2 - 2n + 1
Here we have a formula, 2 n 2 - 2n + 1, for calculating the nth term in the sequence.
Testing The Formula
I can test the formula by using it on previous shapes, which I have built out of the Lego bricks. I will test five of my results. Below are my testing methods and my results.
Single Square
a n 2 + b n + c
2 x 1 2 - 2 x 1 + 1
Answer: 1
Status: Correct
5x5 Cross-shape
a n 2 + b n + c
2 x 4 2 - 2 x 4 + 1
Answer: 25
Status: Correct
x1 Cross-shape
a n 2 + b n + c
2 x 2 2 - 2 x 2 + 1
Answer: 5
Status: Correct
7x7 Cross-shape
a n 2 + b n + c
2 x 5 2 - 2 x 5 + 1
Answer: 41
Status: Correct
3x3 Cross-shape
a n 2 + b n + c
2 x 3 2 - 2 x 3 + 1
Answer: 13
Status: Correct
9x9 Cross-shape
a n 2 + b n + c
2 x 6 2 - 2 x 6 + 1
Answer: 61
Status: Correct
So far, all of the Cross-shapes that I have tested, have been correct. This shows that the formula works for these shapes. I can't, however, prove that it will work for any shape, though by testing some different sized cross-shapes, I can discover how many total squares they consist of and then draw them out, to see if the formula was correct. This will still not prove that it works for any number, however, by extending my two dimensional investigation, I will whether is more reliable. This will go further to proving that the investigation works with any sized shape, though will not make it certain.
Below are two other cross shapes. They are supposed to be 85 and 113 squares. I will now draw them out to test that this is correct.
An 11x11 Cross-shape
a n 2 + b n + c
2 x 7 2 - 2 x 7 + 1
Answer: 85
Counted Answer: 85
Status: Both Correct
A 13x13 Cross-shape
a n 2 + b n + c
2 x 8 2 - 2 x 8 + 1
Formula Answer: 113
Counted Answer: 113
Status: Both Correct
Summary
There is no way to make sure that the formula will work for every sized shape but I tried to make it more reliable by showing that it works with all the ones tested and two other cross-shapes that I had not investigated before, which were all correct.
The only way to be certain that the formula works with every sized cross shape is by testing it on every possible sized one, which would be very difficult. This is why I feel that I have generated enough information to show that the formula that I have found out, 2 n 2 - 2 n + 1, gives the expected results.
