Cubes and Cuboids Investigation.

Authors Avatar

Cubes and Cuboids Investigation

          I am going to investigate the different patterns that occur with different cubes when all the faces are painted of a large cube and then that is separated into smaller cubes and then how many faces of each small cube are still painted. Here are my cubes. They are 2*2*2, 3*3*3 and 4*4*4.

                   

             

          I am going to establish the patterns that recur as the cube gets larger. For example the number of cubes with one face painted, with two faces painted, with three faces painted and the number of cubes with no faces painted when the larger cube is split up. Here is a table:

          Immediately I noticed that all of the cubes have 8 cubes with 3 different faces painted when they are separated. All of these 8 are the vertices of the cube and so every cube except that which has a length of one will have 8 cubes with three faces painted.

          This can be shown in the  table:

         

           Y=8

'font-size:14.0pt; '>The above tells us how many cubes will have three painted faces to find out how many will have two, here is a table:

  Y=12(X-2)

'font-size:14.0pt; '>I noticed this formula because as the differences are 12 then the formula must have something to do with 12X. Also if you look at the diagram above you can see that all the cubes with two painted (brown) are one in from either side and so the formula must have X-2 in it which is the cube length minus 2. Therefore the formula is Y=12(X-2).

'font-size:14.0pt; '>Now I have found out the formula for cubes with 3 or 2 faces painted.

'font-size:14.0pt; '>Here is the table showing the cubes with one painted face:

'font-size:14.0pt; '>Y=(X-2)2*6

'font-size:14.0pt; '>Again we can say that because we want only the cubes with one painted face which are all in the middle (green) it is going to have X-2 in the formula. We can also say that as all of the shapes are cubes, the cubes with one painted face are always going to be in a square so we can say that the formula so far is (X-2)2. Lastly as there are 6 faces on a cube there is going to be 6 similar squares with the same amount of cubes with one face painted and so the formula for cubes with only one painted face is Y=(X-2)2*6.

'font-size:14.0pt; '>The last formula that must be found out to make the set complete is that which tells us the number of cubes with no painted faces.

'font-size:14.0pt; '>Here is a table to show the number of cubes with no painted faces:

'font-size:14.0pt; '>Y=(X-2)3

'font-size:14.0pt; '>Again we can say, as none of the cubes in question are on the outside of the cube and so you have to minus 1 from each side of the length, that X-2 is going to be in the formula. We also know that the cube or cubes that have no painted faces, are embedded around the shell of the large cube and so the cubes in the middle will also form a cube, therefore we can say that the formula is going to be Y=(X-2)3.

Join now!

'font-size:14.0pt; '>Here is a table showing all of the formulae that go together to show how the different number of faces painted on the small cubes are arranged:

'font-size:14.0pt; '>It is possible to check that these formulae work because if they are all added up then it should be equal to the total number of small cubes in the large cube. Which has the formula Y=X3

'font-size:14.0pt; '>Here are the added formulae:

'font-size:14.0pt; '>Y=X3-6X2+12X-8+6X2-24X+24+12X-24+8

'font-size:14.0pt; '>Y=X3

'font-size:14.0pt; '>Therefore my formulae must be correct because when added together they equal Y=X3.

'font-size:14.0pt; '>All of the above covers how to find, how ...

This is a preview of the whole essay