4 28
If I half the 2nd difference then I get 2. I am going to try my formula:
B= 2N2
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
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.
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
work up to more complex shapes.
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.
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
My formula was correct!
3-d Cuboids
I will now do the investigation into cuboid designs.
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.
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:
2x25=50
My formula was correct!
3-d Pyramids
Next I will try to find a formula for 3-d pyramids.
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:
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
286 = 1 1/3 x 62 –6
3
My formula appears to be right.
More 3-d Pyramids
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:
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
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
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
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.
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:
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)
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:
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:
Instead of this
(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:
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:
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:
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.
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
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.
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:
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:
If I were to extend this investigation I would try looking at other shapes, as well as cubes e.g. Pyramids, Prisms etc.
