Results
The nth term of the sequence is 3n – 2 = Hidden face. I have done 7 cubes, now with this formula I can find out how many hidden faces there are on 8 cubes.
3n – 2 = Hidden faces
3x (8) – 2
24 – 2
= 22
This shows us that there will be 22 hidden faces on 8 cubes.
Conclusion
My results show that my prediction was correct. You do need to add 3 each time a cube is added and the formula for this is 3n – 2 = Hidden faces.
I am now going to investigate the number of hidden faces in a rectangular cuboid. I am going to start with 2 cubes joined together.
These 2 cubes have a total of 4
hidden faces. Two of the hidden
faces are where the cubes are joined
together and the other two are underneath
where they are facing down onto the table.
4 faces hidden
8 faces showing
12 faces altogether
Now that that I have counted how many hidden faces there are on 2 cubes I will add 2 more to this and find out how many hidden faces there are on 4 cubes, in a rectangle.
These 4 cubes have a total of 12 hidden faces.
Eight of the hidden faces are where they are
all joined together, the other 4 are underneath
where they are facing down onto the table.
12 faces hidden
12 faces showing
24 faces altogether
I have found out that the number of hidden faces goes up in 8’s so my first formula won’t work with this part. There is another formula that I have found that works with this part, it is 8n – 4.
I predict that 14 cubes will have 52 hidden faces because 8 x 7 = 56 and you then use the formula which is 8n – 4 and 56 – 4 = 52 so that’s how I get my prediction.
I am now going to investigate a 30 cube cuboid. I will find out the number of hidden faces then count the hidden faces row by row, by doing this I will be able to work out the formula for 30 cubes.
During my investigation I noticed that 1 cube has 6 faces on it, I used 30 cubes, so I did the sum 6 x 30 which = 180 then I counted the faces I could see which was 47 so then I took 47 away from 180 which equals 133.
This is the formula you need to use for 30 cubes.
6 (30) – 47 = 133
I have now tried 30 cubes on its side like this.
I counted the showing faces there were 56 faces showing. Now I need to find out how many faces are hidden. I will do this by doing this sum below.
6 (30) – 56
= 124 hidden faces
When you turn the block of 30 cubes on its side you do not get the same amount of hidden faces as the one on its original place.
Overall Conclusion
By looking at my previous investigation looking at the amount of hidden faces in cuboids, I have came across many different formulas and patterns for the first investigation, I used one row of cubes I found a pattern and worked out the formula to be 3n – 2. then I went onto my second investigation I used two rows going up joining together I gathered some results and put it into a table I tried my first formula on the second lot of results but it didn’t work. I found the formula to be 8n – 4. I found the second part much more complicated than the first part. Then I went onto a 30 cube cuboid I found this very hard to work out but I realised the formula was right in front of me I realised there was 6 faces to a cube and I was using 30 cubes so I done 6 x 30 which worked out to be 180 then I counted the number of faces showing which was 47 I subtracted this from 180 to be 133 used another method and I came out with the same answer.