3 joints = 8
4 Joints
PREDICTION: I predict that the first cube will have 0 but as I am increasing
My cubes 1 cm2 each time I think it will increase each time
On a 2 x 2 x 2 cube there are 12 joints each side is made up of four smaller squares where these join at the edges there will always be a four joint this is the same with the other cubes just they will increase with the size of the cube there is a trend for example a 4 x 4 x 4 will be 3 along the edge of the cube its one less than the size of a cube it could be written like this n – 1 with this information I can work out the formulae.
50 x 50 x 50 cube = 588 3 Joints
N x N x N cube = 12 (edges) (n [size of cube] – 1)
Formulae
4 Joints = 12 (n – 1)
5 Joints
PREDICTION: I predict that there will be 0 on the first cube the other will increase with the size of the cube
On a 2 x 2 x 2 cube there are 5 joints in the middle of each side of the cube and one in the centre obviously there are always 6 sides to a cube so, so there are 6 5 joints, it will increase with the larger cubes. With this information I can work out the formulae
50 x 50 x 50 cubes = 14700 5 joints
N x N x N cube = 6(n – 49)² 5 joints
Formulae
5 Joints = 6 (n – 1) ²
6 Joints
PREDICTION: I predict that the 1 x 1 x 1 cube will have 0 6 joints but the other cubes amount of 6 joints will depended on how many layers the cube has and the area of the sides too, but there will not be one upon the edges
On a 2 x 2 x 2 cube there is one six joint a 2 x 2 x 2 is kind of mind up of 8 a 1 x 1 x 1 cube. You could say where these meet in the centre that is where the six rods make the 6 joints. With this information I can work out the formulae
50 x 50 x 50 cubes = 117649
N x N x N cubes = (n – 1)³ 6 joints
Formulae
6 Joints = (n – 1) 3
Cubes: Rods
The next part of the investigation is to find out how many rods in each cube
1 x 1 x 1 Cube
This cube has 12 rods. I know this because each cube has six sides and the 1cm³ cube had 6 1cm square each square is made up of 4 rods, but some share a rod
6 x 2 = 12
2 x 2 x 2 Cube
This cube has 54 rods in total. I know this because there are 24 1cm square on the sides of the cube 24 x 2 = 48 rods the 6 other rods are inside connecting the sides together 24 x 2 + 6 = 54
3 x 3 x 3 Cube
This Cube houses 108 rods. Because there are 54 small squares with some that share rods on the cube 54 x 2 = 108
4 x 4 x 4 Cube
This cube has 300 rods. This is because there are 96 squares that make the faces of this cube with 2 rods each. 96 multiplied by 2 equal 192. Inside the cube there is another, which is the size equal to a 3 x 3 x 3 cube, which has 108 rods. 192 + 108 equals 300
5 x 5 x 5
This cube has 540 rods. This is because there are 60 rods on one face. 60 multiplied by the 6 faces that are on a cube. 60 x 6 equal 360. There are another 6 faces passing the opposite way with 30 exclusive rods. 30 multiplied by 6 equal 180. 360 +180 = 540.
Rods Formulae
From conducting the investigation I could see that the number of rods was equal to the number of 3 joints times 3 add the number of 4 joints times 4 add the number of 5 joints times 5 add the number of 6 joints times 6. That is not all though, as by doing that each rod was counted twice, therefore after doing the above I will divide the final result by 2. This can be put in a formula by multiplying in the relevant numbers into the formulae that I already have and adding these formulae together as shown below:
(8x3) + 4x12(n-1) + 5x6(n-1) 2 + 6(n-1) 3
2
= 24 + 48(n-1) + 30 (n-1) 2 + 6(n-1) 3
2
= 24 + 48n-48 + (30n-30)(n-1) + (6n-6)(n-1)(n-1)
2
= 24 + 48n-48 + 30n2–60n+30 + (6n2-12n+6) (n-1)
2
= 24 + 48n-48 + 30n2 –60n+30 + 6n3-12n2+6n-6n2+12n-6
2
= 6n3+30n2-12n2-6n2+48n-60n+6n+12n+24-48+30-6
2
= 6n3 + 12n2 +6n
2
= 3n3 + 6n2 +3n
=3n (n2+2n+1)
=3n (n+1)(n+1)
=3n (n+1) 2
Therefore the final formula for the number of cubes = 3n (n+1) 2
Justifying Formulae
I can now use my formulae to predict what the results would be in a 6x6x6 cube. If my formulae are correct then the cube would have 8 3joints, 60 4 joints, 150 5 joints, 125 6 joints and 882 unit rods.
(See Sheet 1)
I can see that my prediction was accurate and consequently my formulae are correct.
To illustrate the increase of joints and the patterns that form I have made a graph
Also a graph to illustrate the increase in rods
Cuboids (Sheet 2)
I started the Investigation by drawing a cuboids 1 x 1 x 2 then 2 x 2 x 4 and then
3 x 3 x 6 and finally 4 x 4 x 8 shape. I thought 4 different sized cuboids would be enough to work out formula and trends that may come up. These are my results for the joints of a cubiod. (note: used cubes findings to support pridctions)
I first counted the joints then rods, and then using my results created these formulae showing how to work out the number of joints in each cube.
3 Joints
PREDICTION: I predict that each of the measured cuboids will have 8 3 joints
There is always 8 3 joints because each corner has a 3 joint and no where else upon the shape any size cube will have 8 3 joints
50 x 50 x 100 cubiod = 8 3 joints
N x N X N cubiod = 8 3 joints
Formulae
3 joints = 8
4 Joints
PREDICTION: I predict that each cuboid will have 4 4 joints the other cubes will increase by a certain constant number at a time
There are 4 4 joints on a cuboid. this
There are always 4 four joints on a 1 x 1 x 2 cuboid as the cuboid is made by joining two cubes. Where they join, on the four corners, there will always be four 4 joints. This will be the same with all cuboids, although the numbers will be increased each time. With this information I can work out:
50 x 50 x 100 cuboid = 784 4 joints.
N x N x N cuboid = 8(n – 1) + 4(2n – 1) joints.
Formulae
4 Joints = 8(n – 1) + 4(2n – 1).
5 Joints
PREDICTION: I predict that the 1 x 1 x 2 Cubiod will have 5 joints but the other cuboids the number of 5 joints will increase related to the longer side of the cubiod
There are always 14 five joints on a 2 x 2 x 4 cuboid as each the 2 x 2 face is made of four squares. Where they meet in the centre creates a 5 joint. There are two 2 x 2 faces to the cuboid, so there are 2 five joints. The larger face, 2 x 4, has 3 five joints. There are four of these sides. 4 multiplied by 3 equals 12. 12 plus 2 equals 14 five joints on the cuboid.
A 50 x 50 x 100 cuboid = 240,296 five joints.
A n x n x n cuboid = 10(n – 1) 2 + 4(n – 1) joints.
Formula
10(n – 1) 2 + 4(n – 1).
6 Joints
PREDICTION: I predict that the first cuboid will have no 6 joints, but on the other cuboids the number of 6 joints will increase as the size of the cuboid increases.
There are always 3 six joints on a 2 x 2 x 4 cuboid. The cuboid is made up of 2, 2 x 2 x 2 cubes, which each have 1 six joint in the centre. There is also one six joint holding the two together. With this information I can work out that:
100 x 100 x 200 cuboid = 240,099 six joints.
N x N x N cuboid = (n – 1) 2 x (2n – 1) joints.
Formula
6 joints = (n – 1) 2 x (2n – 1).
RODS
The next thing I needed to do was determine how many rods were in each cuboid. I did this in the same way as the cubes.
1 x 1 x 2 CUBOID
This cuboid has 20 rods. This is because it is made up of 2 1 x 1 x 1 cubes. These cubes are made up of 12 rods. 12 x 2 equals 24. As the cubes join in the centre there are four less rods. 24 minus 4 equals 20 rods.
2 x 2 x 4 CUBOID
This cuboid has 96 rods. This is because the two 2 x 2 x 2 cubes that it is made from have 54 rods each. Where they meet at the centre there are 12 less rods, as they makes up 2 faces. 108 minus 12 equals 96.
3 x 3 x 6 CUBOID
This cuboid has 264 rods. This is because the smaller face has 24 rods. There are 7 faces through the whole cuboid. 24 multiplied by 7 equals 168. Each larger face has 54 rods. As the vertical rods have been counted through the smaller face, only the horizontal rods count. There are 24 horizontal rods, and 4 larger faces through the cuboid. 24 multiplied by 4 equals 96. 96 plus 168 equals 264 rods.
4 x 4 x 8 CUBOID
This cuboid has 560 rods. This is because two 4 x 4 x 4 cubes that make up the cuboid have 300 rods each. The face has 40 rods. Where they meet in the centre the 40 rods make 2 faces, so 40 rods must be taken away. 600 minus 40 equals 560.
With this information I can calculate that a 100 x 100 x 200 cube will have 6,140,000 rods.
An n x n x n cube will have 6n3 + 10n2 + 4n rods.
Formula
Rods = 6n3 + 10n2 + 4n
To illustrate the increase of joints and the patterns that form I have made a graph
Also a graph to illustrate the increase in rods
