# Hidden Faces Of A line Of Cubes

Essay by smelly_grubies April 2004

Hidden Faces

Introduction: I need find out a formula for the number of hidden and visible faces of a line of cubes placed on a flat table. E.g. for a line of three cubes there is a total of 18 faces, 11 of them are visible and 7 of them are hidden. I will need to find a formula for a quicker way of working this out. I will draw the cubes and write up tables to help me to find the formula. I think that this is a good method because it's easy to see.

I have found out a pattern. Multiply the middle cubes in each line buy three and then add eight to find out the number of visible faces e.g. for four cubes there are two middle cubes, so 2x3= 6 and then add on eight, equalling fourteen.

I will now start putting my cubes into a table.

This will make it easier for me to start looking for more patterns in order to find formula.

Number of cubesHidden faces Visible facesTotal faces

1 1 5 6

2 4 8 12

3 7 11 18

4 10 14 24

5 13 17 30

6 16 20 36

7 19 23 42

8 22 26 48

I have predicted a formula for finding the visible faces:

Number of cubes - 2 x 3 + 8 = Visible faces

I will now test my formula:

1 - 2 x 3 + 8 = 5 CORRECT

2 - 2 x 3 + 8 = 8 CORRECT

6 - 2 x 3 + 8 = 20CORRECT

Algebraic form:

v= 3 (n - 2) + 8

My formula is correct. I devised the formula from the first pattern and I will now see if I can look for...