# The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows.

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Introduction

MATHS COURSEWORK HIDDEN FACES. Aim: The aim of this coursework is to find a global formula for the total number of hidden faces for any number of cubes in rows. A cube has six faces in total. Hidden faces are faces that cannot be seen when a cube is placed on a table or in rows along side other cubes. If you place five cubes along side each other on to a table, they have a total of 30 faces of which 13 faces are hidden and 17 can be seen. In order to find the global formula I will have to find general formulae for the different number of rows by producing tables and drawing diagrams. I will first find out a general formula for one row of cubes. I will start at one cube and go up to eight cubes in a row. Results: Cubes in a row Total faces Faces seen Faces hidden 1x1 6 5 1 1x2 12 8 4 1x3 18 11 7 1x4 24 14 10 1x5 30 17 13 1x6 36 20 16 1x7 42 23 19 1x8 48 26 22 6n 3n+2 3n-2 In the table and graph above I have shown the relationship between the cubes, the total number of faces, their hidden faces and the faces that can be seen. ...read more.

Middle

I will again do another table to find the general formula. Number Of Cubes. 1 2 3 4 5 6 7 8 Hidden Faces. 4 12 20 28 36 44 52 60 1st Difference + 8 + 8 + 8 + 8 As you can see from the table above there is still one line of difference so therefore it is still a linear equation: y=mx+c The linear rule is: tn=an+c In the above equation tn is the number of hidden faces and n is the number of cubes. a being the first line of difference, it is 8. Using the same method as before I found out the value of c. tn=an+c 28=8(4)+c 28=32+c Rearranged: 28=32+c 28?32=c 28?32= ?4 c= ?4 I now know the value of c which is ?4. By putting this into an equation I get: tn=8n?4 I am now going to test the equation to see if it is right or wrong. tn=8n?4 tn=8(6) ?4 tn=48?4 tn=44 The formula was correct and the number of hidden faces (tn) was the same as in my results. The next part of this coursework I am going to find out the general formula for the number of hidden faces in three rows of cubes. ...read more.

Conclusion

This gives you the final equation of: tn=13n?6. To prove that this equation is correct and works, I need to first put the n back into the equation with a number of cubes from the table. If this equation is correct then the answer that I will get should be the same as number of hidden faces. tn= an+c tn= 13n?6 tn= 13(2)?6 tn= 26?6 tn= 20 In the above equation I decided to use two cubes, as you can see for two cubes the number of hidden faces is 20, which is the answer I got from my equation. This tells us that the formula I got for the total number of hidden faces is correct. GLOBAL FORMULA. A global formula is a formula that works on all the rows and cuboids made up of cubes. The first thing I did was find the equations for the width, height and length. These were as follows: Width=w Length=l Height=h To find width the equation is as follows: W=lw(2l-1)hw To find length the equation is as follows: L=2l(w-1)h To find height the equation is as follows: H=2l(h-1)w The addition of these three equations results in the global formula. I then simplified this and it was as follows: 6wlh-2(hw+hl)-lw ...read more.

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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