# Cube Shaped Boxes and Supermarket Displays

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Cube Shaped Boxes and Supermarket Displays

Introduction

The question:

Boxes made in the shape of a cube are easy to stack to make displays
in supermarkets.

Investigate!

Plan
====

I will carry out this investigation by following these points:

1. Simplify the question by using 2-d shapes.

2. Draw 2-d designs.

3. Draw 3-d designs.

4. Evaluate my work.

Detailed Plan

To investigate each shape I will follow a pattern:

1. I will state which shape I am investigating.

2. Draw the shape or a bird's eye view to show the shape.

3. Draw a difference table to show whether it is a linear or quadratic
formula.

4. Make a guess as to what the formula might be.

5. Check it. If it is right prove it by using the next shape in the
series. If it is not right work out the next part of the formula.

2-d Squares and Rectangles

[IMAGE]To simplify this investigation I will start off with an easy
shape and work my way onto more complex shapes.

[IMAGE]

[IMAGE]

[IMAGE]

To find a formula for this pattern I will draw a difference table.

Number of Layers (N) Number of boxes (B) 1st Difference (1st)

1 2
2
2 4
2
3 6
2
4 8

This shows that each time you add a layer you add two more boxes. I
will try and find a formula for this by multiplying the difference by
the length of the side.

N x 1st =

2

4

6

8

B=

2

4

6

8

As you can see the numbers are the same, meaning the formula for this
pattern is:

B=2N

I shall prove his formula to be right by trying the next shape in the
sequence.

MLA Citation:
"Cube Shaped Boxes and Supermarket Displays." 123HelpMe.com. 05 Dec 2019
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### Popular Essays

[IMAGE] 2x5=10

As you can see my formula was correct!

2-d Triangles

[IMAGE]I shall try another shape:

[IMAGE]

[IMAGE]

[IMAGE]

[IMAGE]

I will find a formula for this pattern by, first of all, drawing a
difference table:

Number of Layers (N) Number Of Boxes (B) 1st difference 2nd Difference

1 1

2

2 3 1

3

3 6 1

4

4 10 1

5

5 15

As this table goes into a second difference this means it is a
quadratic equation, implying the highest power of N is N2. I am going
to try halving N2.

B=
--

1

3

6

10

15

½ N2=

0.5

2

4.5

8

12.5

Whilst trying to find out the formula I noticed that to make B from ½
N2 all you need to do is add ½ L. So the equation for this shape is:

B= ½ N2 + ½ N

To check this formula I will try it on the next shape in the series:

[IMAGE]

B= 0.5 x 62 +0.5 x 6

B=21

My formula is correct.

2-d Pyramids

[IMAGE]Next I will try a more complex shape:

[IMAGE]

[IMAGE]

[IMAGE]

I will find a formula for this pattern by, first of all, drawing a
difference table:

Number of layers (N) Number of boxes (B) 1st difference 2nd difference

1 1

3

2 4 2

5

3 9 2

7

4 16

As this table goes into a second difference this means it is a
quadratic equation, implying the highest power of N is N2. I think
that if I square the number of layers I will get the number of boxes.

N2 =

1

4

9

16

B=

1

4

9

16

As you can see the results are equal meaning the formula for this
pattern is:

B=N2

I shall prove this formula by trying the next shape in the sequence:

[IMAGE]

52=25

As you can see my formula was correct!

More 2-d Pyramids

Next I will find a formula for a different pyramid:

[IMAGE][IMAGE][IMAGE][IMAGE]

To find a formula I will draw a difference table:

Number of layers (N) Number of boxes (B) 1st difference 2nd difference

1 1

2

2 3 1

3

3 6 1

4

4 10 1

5

5 15

I have noticed that this shape has the same difference table as the
triangular stacking pattern. I am guessing that this means they have
the same formula. I will check my guess by trying the formula on the
next shape in the series.

[IMAGE]

B= 0.5 x 62 +0.5 x 6

B=21

My formula is correct.

During the two dimensional patterns I have approximated the formulae
and made educated guesses as to what the relationship between the two
variables is. Now I am going to try and find a relationship that links
the formulae and produce a final formula to calculate any pattern that
has a repeating difference in the second difference column.

To do this I decided to compare all of the formulas to see if there
are any similarities.

B= N2

B= ½ N2 + ½ N

I will start off looking at the quadratic parts of the formulas.

I have noticed that in all of these formulas there is something x N2.
Next I tried to see if there was a pattern between what N2 was
multiplied by.

1N2

½ N2

I think that it is ½ of the second difference multiplied by N2.

Next I will see if there is a pattern between the linear parts of the
formulas.

After extensive calculating and research I have come to the conclusion
that there is no pattern between the linear sections of the formulas.

The formula is then:

½ 2nd difference x N2 + the remaining difference.

This then helps to work out the formula, as all you need to do is work
out the linear section.

[IMAGE]To prove this I will test it on a new shape.

[IMAGE]

[IMAGE]

[IMAGE]

I will find a formula for this pattern by drawing a difference table:

Number Of Layers (N) Number of Boxes (B) 1st Difference 2nd Difference

1 1

5

2 6 4

9

3 15 4

13

4 28

If I half the 2nd difference then I get 2. I am going to try my
formula:

B= 2N2
------

2N2 =

2

8

18

32

B=

1

6

15

28

As you can see the numbers do not match which means I need to add
something else to the formula. I have looked at the numbers and
realised that all you need to do is take away N. I then get this:

B= 2N2 -N

2N2 - N=

1

6

15

28

B=

1

6

15

28

As you can see the numbers are the same meaning my formula for this
pattern and my overall formulas are right. I shall prove the formula
for this pattern by trying it on the next shape in the series.

[IMAGE]

2 x 52 - 5 =45

My formula works.

3-D Cubes

Next I will try 3-d shapes. I will start with another easy one and

[IMAGE]work up to more complex shapes.

[IMAGE]

[IMAGE]

[IMAGE]

[IMAGE]

I will find a formula for this pattern by drawing a difference table:

Number of Layers (N) Number of Boxes (B) 1st difference 2nd
differerence 3rd differerce

1 1

7

2 8 12

19 6

3 27 18

37 6

4 64 24

61

5 125

From looking at this diagram I shall try to find a formula by cubing

the number of layers.

N3=

1

8

27

64

125

B=
--

1

8

27

64

125

As you can see the results are the same meaning the formula for this
sequence is:

B = N3

I will prove this by trying it for the next shape in this series:

63 =216

[IMAGE]My formula was correct!

3-d Cuboids

I will now do the investigation into cuboid designs.[IMAGE]

[IMAGE]

[IMAGE]

[IMAGE]

I will find a formula for this pattern by drawing a difference table:

Length of side (N) Number of boxes (B) 1~ difference 2~ difference

1 2
6
2 8 4
10
3 18 4
14
4 32

Next I will try to find a formula for this pattern by using the
difference table.

I think that if you square the number of layers then multiply it by
two you should get the number of boxes.

2N2 =
-----

2

8

18

32

B=

2

8

18

32

As you can see the numbers match, meaning the formula for this pattern
is:

B=2N2

I will test this formula by trying it for the next shape in the
series:

[IMAGE]2x25=50

My formula was correct!

3-d Pyramids

[IMAGE]Next I will try to find a formula for 3-d pyramids.

[IMAGE]

[IMAGE]

[IMAGE]

[IMAGE]

To find a formula I will now draw a difference table:

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
9
2 10 16
25 8
3 35 24
49 8
4 84 32
81
5 165

I have noticed that them is a pattern between the number of boxes on a
layer and the layer that they are on. I have put this pattern of
numbers into a table:

Layer

1

2

3

4

5

Number Of Boxes

1

10

35

84

165

Pattern

1x1

1x1+3x3

1x1+3x3+5x5

1x1+3x3+5x5+7x7

1x1+3x3+5x5+7x7+9x9

As there is a third difference this means that N will be cubed. To
find out the relationship between the number in the third difference
and the number that I multiply N3 by I will first test 2N3 then 3N3.

Number Of Layers 2N3 1st Difference 2nd Difference 3rd Difference

1 2

14

2 16 24

38 12

3 54 36

74 12

4 128 48

122

5 250

When N3 is multiplied by two the third difference has a constant of

12. The relationship between 2 and 12 is either multiplying by six or
adding twelve. I will see which is right by drawing a difference table
for 3N3 .

Number Of Layers 2N3 1st Difference 2nd Difference 3rd Difference

1 3

21

2 24 36

57 18

3 81 54

111 18

4 192 72

183

5 375

As you can see the number in the third difference column is 18. This
is 3 x 6. This shows me that the relationship between the number that
is multiplied by N3 and by the number in the third difference column
is:

The number in the 3rd difference column

6

Using this formula I can now work out the formula for my first 3-d
pyramid.

8/6 = 1 1/3

This shows me that the first part of the formula should be:

1 1/3 N3

To find the next part of the formula I will use the formula above and
use it to look at the difference between the answers from 1 1/3 N3
multiplied by the layers.

No. Of Layers No. Of Boxes 1 1/3 N3 1st Difference

1 1 1 1/3 1/3

2 10 10 2/3 2/3

3 35 36 3/3

4 84 85 1/3 4/3

5 165 166 2/3 5/3

To find the number of boxes using the formula 11/3 N3 you have to add
a fraction and the numerator of the fraction is equal to the number of
layers. This means that the overall formula for this stacking design
is:

B= 1 1/3 N3 -N

3

To prove this formula I will try it on the next shape in the series:

The first thing I figured out was if you do N x odd numbers2 then you
get the total. For the six layered pyramid it would be:

1x1+3x3+5x5+7x7+9x9+11x11 = 286

Or

12 + 32 + 52 + 72 + 92 + 112 = 286

[IMAGE]

286 = 1 1/3 x 62 -6

3

My formula appears to be right.

More 3-d Pyramids

[IMAGE]

This time I am going to draw the bird's eye view of the shapes rather
than the 3-d view.

To find a formula I will now draw a difference table:

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
16
2 17 33
49 18
3 66 51
100 18
4 166 69
169
5 335

Using the number from the 3rd difference column I can figure out the
first part of the formula.

18

6

The answer to this is 3. That means the first part of the formula is:

3N3

To find the rest of the formula I will put this part of the formula
into a table to see what I get:

No. Of Layers (N) 3N3 No of Boxes (B) Difference between 1st
difference 2nd difference

B and 3N3

1 3 1 -2

-5

2 24 17 -7 -3

-8

3 81 66 -15 -3

-11

4 192 166 -26 -3

-13

5 375 335 -40

As there is a repeating 2nd difference I will use the formula that I
worked out earlier.

½ 2nd difference x N2

-3/ 2 = -1 ½

-1 ½ N2

This is the second part of the formula. I will add this to the first
part of the formula to get this:

B = 3N3 -1½ N2

I now need to find out if this is the complete formula or whether I
need to add anything else. To do this I will put this formula into a
table:

No. Of Layers (N) 3N3-1 ½ N2 No of Boxes (B) Difference between 1st
difference

B and 3N3-1 ½ N2

1 1 2/3 1 -1/2

1/2

2 18 17 -2/2 1/2

3 67 2/3 66 -3/2 1/2

4 168 166 -4/2 1/2

5 337 2/3 335 -5/2

Whilst looking at the table I noticed that the numerator in the
fraction for the amount that has to be removed (The column with the
difference between my formula and the total number of boxes) is the
same as the number of layers in the stack. From this I can finish the
formula because the denominator will be constantly two and the
numerator is dependent on the number of layers in the stack. The last
part of the formula will therefore look like this:

- N

2

I then added this to the rest of the formula to get:

B = 3N3 -1½ N2 - N

2

I now think that my formula is correct so I will test it on the next
shape in the series:

B= 3x63 - 1 ½ x 62 - 6

2

B= 591

To test this I will add it onto the first difference table that I made
to see if it carries on the repeating difference of 18:

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
16
2 17 33
49 18
3 66 51
100 18
4 166 69
169 18
5 335 105

256

6 591

3-d Pyramids

[IMAGE]

Again I will draw a bird's eye view of my pattern rather than the 3-d
view.

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
25
2 26 56
81 32
3 107 88
169 32
4 276 120
289
5 565

Using the number in the 3rd difference column I can work out the first
part of the formula.

32

6

This equals 5 1/3 meaning the first part of the formula is:

B= 5 1/3 N3

To find the next part of the formula I will put this into a table:

No. Of Layers (N) 5 1/3 N3 No of Boxes (B) Difference between 1st
difference 2nd difference

B and 5 1/3N3

1 5 1/3 1 -4 1/3

-12 1/3

2 42 2/3 26 -16 2/3 -8

-20 1/3

3 144 107 -37 -8

-28 1/3

4 341 1/3 276 -65 1/3 -8

-36 1/3

5 666 2/3 565 -101 2/3

When I put the first part of the formula into a table I found that it
went into the second difference meaning I should use the formula I
found earlier for any pattern with a repeating second difference.

½ 2nd difference x N2

-8

2

-4 N2

I will add this onto the formula then see if I need to add anything
else to the formula.

B= 5 1/3 N3 -4 N2

No. Of Layers (N) 5 1/3 N3- 4N2 No of Boxes (B) Difference between 1st
difference

B and 5 1/3 N3- 4N2

1 1 1/3 1 -1/3

1/3

2 26 2/3 26 -2/3 1/3

3 108 107 -3/3 1/3

4 277 1/3 276 -4/3 1/3

5 566 2/3 565 -5/3

Like in one of my previous formulas the number of layers is equal to
the numerator of the fraction that has to be taken away from the
formula. Using this information I can complete the formula which when
completed will look like this:

B= 5 1/3 N3 -4 N2 -N

3

To check my formula I will test it on the next shape in the series:

B= 5 1/3 x 63 -4 x 62 -6

3

B= 1006

To test this I will add it onto the first difference table that I made
to see if it carries on the repeating difference of 32:

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
25
2 26 56
81 32
3 107 88
169 32
4 276 120
289 32
5 565 152

441

6 1006

My formula appears to be right.

3-d Pyramids

[IMAGE]

The last formula for 3D pyramids that I am going to try and
investigate will be a pyramid with only a half box step.

Number of Layers (N) Number of boxes (B) 1st difference 2nd difference
3rd difference

1 1
4
2 5 5
9 2
3 14 7
16 2
4 30 9
25 5 55

Using the number in the third difference column I can find the first
part of the formula:

2

6

The answer to this is 1/3. This means the first part of the formula
is:

1/3 N3

I will now put this into a table to find the next part of the formula:

No. Of Layers (N) 1/3 N3 No of Boxes (B) Difference between 1st
difference 2nd difference

B and 5 1/3N3

1 1/3 1 2/3

1 2/3

2 2 2/3 5 2 1/3 1

2 2/3

3 9 14 5 1

3 2/3

4 21 1/3 30 8 2/3 1

4 2/3

5 41 2/3 55 13 1/3

Again the table goes into a second difference meaning I can use the
formula I found earlier:

½ 2nd difference x N2

½ N2

I will now add this to the rest of my formula:

B= 1/3 N3 + ½ N2

I will now see if my formula is complete or if I need to add anything
else:

No. Of Layers (N) 1/3 N3 + ½ N2 No of Boxes (B) Difference between 1st
difference

B and 1/3 N3 + ½ N2

1 5/6 1 1/6

1/6

2 4 2/3 5 2/6 1/6

3 13 1/2 14 3/6 1/6

4 29 1/3 30 4/6 1/6

5 54 1/6 55 5/6

Each time the layers increase the amount of sixths added on increases
with, it at the same rate. This is because the numerator is equal to
the number of layers in the stack. From this I will finish the
formula:

B = 1/3 N3 + ½ N2 + N

6

To test this pyramid I will draw the sixth layer for this series and
count the total amount of boxes and compare this to my formula.

B = 1/3 N3 + ½ N2 + N

6

B = 1/3 216 + ½ 36 +6

6

B = 72 + 18 + 1

B= 91

[IMAGE]

My formula appears to be correct.

I now have 4 formulas for 3-d Pyramids. I am going to try and see if
there is a formula that links all four 3-d square-based pyramids.

The first thing that I am going to do will be to put the formulas
together and find a pattern, once I have found a pattern I can then
predict a fifth formula and check it. Then I will try and find the
formula. Here are the formulas that I have found so far:

B= 1 1/3 N3 -N

3

B = 3N3 -1½ N2 - N

2

B= 5 1/3 N3 -4 N2 -N

3

B = 1/3 N3 + ½ N2 + N

6

To start off I am going to concentrate on the first part of the
formula, the N3 part.

A= the length of the step.

[IMAGE]

Previously when I had a repeating second difference I would half it,
so I thought that this would be a good start for this formula. I also
used the difference table to work out the first part of the formula
for a pyramid with A as 2 ½ boxes.

Half of two thirds is one third so the beginning of the formula would
have to have a square in it and because I am working out a formula for
the formulas N does not change but A does so I will use A instead of
N. Also A changes when the pyramids change N does not. The first part
of the formula would be something like:

1/3 A2

I will put this into a table to see what I have gained:

[IMAGE]

I tried experimenting with this formula but this formula does not seem
to be correct. Next I am going to try making a formula using 1,2,3,4,5
instead of ½,1,1½, 2, 2½ because it is easier to spot patterns between
1,2,3,4,5 than ½,1,1½, 2, 2½. To do this I must multiply A by 2. This
changes the formula to:

1/3 (2A2)

[IMAGE]
Next I will try this in a table:

If you double my formula then you get N1/3 so the formula for the
first part of the formulas is:

(1/3(2A2)) x 2

To test this I will use one of the formulas that I have already
discovered and see if the formula comes up with the same start as the
proper formula. The formula that I will use will be A=1½, the number
that N3 should be multiplied by is three.

(1/3(2A2)) x 2

(1/3(4½)) x 2

(1 ½) x 2

The number that N3 is multiplied by is 3

The formula is correct I just have to add N3 into it:

((1/3(2A2)) x 2) N3

Now that I have found the first part of the formula I will now
investigate the second part.

I will start by drawing a table of all of the second parts of the
formulas:

[IMAGE]

From this table I am then going to half the second difference column
in order to find out part of the sum.

-1 x ½ = -½

I am now going to use the other formula for the first part of the
formulas as a guide for this formula. In my other formula I had a
third instead of minus a half but I still had A2 which is a piece of
the second part of the formula. If I put this formula together in the
same way I did the last formula I get:

(1/3(2A2))

You get this

(-1/2(2A2))

I have left the last part out on purpose because I only want to
concentrate on the beginning piece first and not on the x2 part. If
you put this into a table you get:

[IMAGE]
I cannot see a useful pattern in this table so I will change the
formula. I am going to experiment by changing parts in my formula
until I get close to the right formula. I am going to start by
changing the A2 to outside the brackets:

(-1/2(2A)2)

This formula has a different outcome to the other formula so I will
enter the results from it into a table:

[IMAGE]

With this formula the difference between N2 and (-1/2(2A)2) is double
the amount of A so if I change my formula to include + 2A at the end
it should be correct.

((-1/2(2A)2) + 2A)N2

I will test this formula by using it to work out a shape where A = 3
and then use a difference table to confirm it.

((-1/2(2A)2) + 2A) N2

(-1/2(36) + 2A) N2

(-18 + 2A) N2

-12 N2

Now I will draw the difference table to prove the formula:

[IMAGE]

From this table I can deduce that my formula is correct and when added
onto the first part of the formula you get:

B= ((1/3(2A2)) x 2) N3 + ((-1/2(2A)2) + 2A) N2

Now I must investigate the last part of the formula, N.

6

In all the previous number patterns that I have studied there has been
either a steady increase or decrease following a line but with this
pattern the numbers are following a curve. If this is true then 1½
would be the center point and therefore the translation would be 11/2.

All the formulas can be expanded so that they all have six as the
denominator, so this is what I will use while doing my investigation.

[IMAGE]

To calculate the formula I am going to put all the endings of the
formulas into a table:

As there is a second difference column and all the numbers in it are
the same this means that I must half 2/6:

1/6 N2

Instead of using N I am going to use A because A changes with each
pyramid but N doesn't. Here is an example of A instead of N.

1/6 A2

In all of the previous parts of the formula A has been multiplied by
two so I will apply the same rule to this part e.g.

1/6 (2A)2

[IMAGE]
I am going to put this formula into a table and compare the results to
N/6.

From this table I have seen a pattern that links 1/6 (2A)2 and N/6.
The pattern that I have spotted is subtracting one with the addition
of ½ A. To work out the rest of the formula I am going to look at the
relationship between A and the difference between 1/6 (2A)2 and N/6.
To do this I am going to put the data in a table.

[IMAGE]

I am first going to look at the first part of the table where A = ½.
To get zero you have to subtract a half, if you apply this to all of
the values for A you get:

[IMAGE]

I am now going to compare A-½ and the difference between 1/6 (2A)2 and
N/6. Looking at the table I have realised that A- ½ equals the
difference between 1/6 (2A)2 and N/6 multiplied by two. I am now going
to put this into a formula:

-((A-½) x 2)

The next thing that I am going to do is to add this part of the
formula to the other part of the formula that I worked out earlier:

(1/6 (2A)2)-((A-½) x 2)

Now I have to add N to the formula to make the last part of the
formula complete and as N is always the numerator and the denominator
can always be six the complete formula will look like this.

((1/6 (2A)2)-((A-½) x 2))N

6

I am now going to prove this formula by testing it with A as 2½.

((1/6 (2x2½)2)-((2½-½) x 2))N

6

((1/6 (5)2)-(2 x 2))N

6

((1/6 25)-4)N

6

(41/6-4)N

6

1/6N

6

The correct answer should be 1 N but it is one sixth instead.

6

To correct this you must multiply the top half of the formula by six
so that you have whole numbers instead of sixths. The new formula will
look like this:

((1/6 (2A)2)-((A-½) x 2)x6)N

6

I will now prove this by trying it wit another number. I am going to
use A as 1½.

((1/6 (2x1½)2)-((1½-½) x 2)x6)N

6

((1/6 (3)2)-(1 x 2)x6)N

6

(((1/6 9)-2)x6)N

6

((1½-2)x6)N

6

(3/6x6)N

6

This is my prediction -3N

6

The answer is -N that is the same as-3N

2 6

This has proven my formula correct, this means that this is the last
part of my formula. I can now add this onto the rest of my formula and
this will give me the formula to workout the total number of boxes in
any 3D square based pyramid. My complete formula is:

B= ((1/3(2A2)) x 2) N3 + ((-1/2(2A)2) + 2A) N2 +

((1/6 (2A)2)-((A-½) x 2)x6)N

6

I do not have to prove this formula correct because I have been
proving it as I went along. I have already proven that the individual
parts of the formula are correct and this means that when they are put
together they will create the correct formula.

I can now use this formula to workout any formula for any
three-dimensional square-based pyramid and that will give me the
formula for the number of boxes depending on the number of layers.

Conclusion

I have completed the task, which was to investigate cubes and how well
they stack. I have come up with these formulas:

Shape:

Formula:

2-d Rectangles

B=2L

2-d Triangles

B= ½ L2 + ½ L

2-d Pyramids (A=1)

B=L2

2-d Pyramids (A=1/2)

B= ½ L2 + ½ L

2-d Pyramids (A= 2)

B= 2L2
------

3-d Cubes

B = L3

3-d Cuboids

B=2L2

3-d Pyramids (A=1)

B= 1 1/3 N3 -N

3

3-d Pyramids (A =1 ½)

B= 3x63 - 1 ½ x 62 - 6

2

3-d Pyramids (A= 2)

B= 5 1/3 N3 -4 N2 -N

3

3-d Pyramids (A= ½)

B = 1/3 N3 + ½ N2 + N

6

Any Pattern with a repeating 2nd difference

½ 2nd difference x N2 + The remaining difference

Any square based Pyramid

where A= the distance of the step and N = The number of layers.

B= ((1/3(2A2)) x 2) N3 + ((-1/2(2A)2) + 2A) N2 +

((1/6 (2A)2)-((A-½) x 2)x6)N

6

If I were to extend this investigation I would try looking at other
shapes, as well as cubes e.g. Pyramids, Prisms etc.