# The aim of my investigation is based on the number of hidden faces and faces in view of cubes that are placed on a table.

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Introduction

Hidden Faces Coursework Investigation

By Mark Costa

## Introduction

The aim of my investigation is based on the number of hidden faces and faces in view of cubes that are placed on a table. From examining the results of my investigation I will hopefully create a formula for each set of cubes that I exam. The procedure of examining these cubes will be done through drawing 3D pictures of the cubes in their patterns on triangular spotty paper, which I will draw myself. Each set of cubes will contain different patterns that will allow me to exam the cubes in varying scenarios and compare different results and formulas that I will create. Through comparing these scenarios I will then amount to a conclusion that will evaluate what I have covered in my investigation.

## Background Information

I already have obtained some background information from examining the task sheet that is set with the investigation at hand. Each individual cube contains six faces; some may be in view while others will be hidden depending on how the cube is placed. When a cube is placed on a table only five out of six faces are in view, therefore one face is hidden. This simple information could be used to conduct a simple formula:

6(all faces)-(number of viewable faces)=(hidden faces) e.g. 6-5=1 therefore the number of hidden faces would be 1.

When five cubes are lined up together in a row, there is a total of thirty faces as 6x5=30. Seventeen of the faces are viewable in a 360 angle. Thirteen of these faces are hidden; the number of faces hidden confirms my first formula:

30-17=13.

Middle

Total Number Of Faces

Hidden Faces

Faces In View

72

38

34

84

45

39

96

52

44

108 | 59 | 49 |

120 | 66 | 54 |

Both formulas equal to the above results, as does the Lego Models based on the drawn set. My discoveries through studying the first and second formula I have used have shown that a simple formula could be used to work out similar formulas, however I wish to challenge this hypothesis by constructing many different sets of cuboids. This concept leads me to my next set of cuboids.

Third Set Of Cuboids

As in the second set of cuboids, each set increases by two cubes although in this set the cubes are placed side-by-side rather then on top. I believe that changing the shape of the cubes will affect the results on the chart, yet I am not sure if the same formula can be used, as I have not done any cubes side-by-side in any previous cuboids.

I once again used the=(all faces)-(number of viewable faces)=(hidden faces) formula to complete the chart which displayed a pattern in the set. I have learnt through observing the results of the table that the number of faces in view increases by four faces each set of cube, whilst the number of hidden faces increase by eight faces each set of cubes.

As in the first set and second, this set increase’s by a certain amount of cubes constantly; therefore I will be able to apply the first formula I used. This set of cuboids may rise by a certain amount of cubes but it does not rise by the same amount as in the first set, which was three hidden faces each set or as the second, which was by seven hidden faces.

As this set of cubes and the second set both have a congregant appearance they both have the same results in the ‘Total number of faces’ section in the chart, which was that the numbers increase by twelve faces each set of cubes. This is completely unlike the first set, which increased by six. I have come to a conclusion that even though the set of cuboids may have the same number of faces the number of hidden faces and faces in view will differ due to the shape or rotation of the cuboids.

This set of cuboids the faces in view increased by four whilst the number of hidden faces increased by eight faces. The previous set of faces increased by seven hidden faces and five faces in view. There is a one-face increase in the results for hidden faces and one face decreased in the results of faces in view, in this set compared to the second set.

An increase and decrease in faces is due to the position of the shape, due to faces that are facing the base (therefore hidden) and the other faces that are not hidden. In this case in the second set of cuboids because the cubes are facing upright, it starts with only one cube in contact with the base, whilst in this set two cubes are facing the base. Therefore because in this set more cubes are facing the base, then more cubes will be hidden then in the second set.

Here are the formulas for my third set of cubes:

(Each cube added after first cube)+8(towards total of hidden faces)

Or

(Each Face in view)-(total number of faces)=(hidden faces) this formula is based on the original formula I used earlier, which still can be used in this case.

Also

(Number of hidden faces increase in set 2)+1=(hidden faces) this formula is based on the comparison of set 2 and set 3, which could possibly be used in later formulas that are similar in shape.

Here are my future predictions on a chart:

Third Set:

Total Number Of Faces | Hidden Faces | Faces In View |

72 | 44 | 28 |

84 | 52 | 32 |

96 | 60 | 36 |

108 | 68 | 40 |

120 | 76 | 44 |

Compared to Second Set:

Here are my future predictions on a chart:

Total Number Of Faces | Hidden Faces | Faces In View |

72 | 38 | 34 |

84 | 45 | 39 |

96 | 52 | 44 |

108 | 59 | 49 |

120 | 66 | 54 |

Conclusion

Here are my formulas for my forth set of cuboids:

(Each cube added after first cube)+18(towards total of hidden faces)

Or

(Each Face in view)-(total number of faces)=(hidden faces) this formula is based on the original formula I used earlier, which still can be used in this case.

I learnt that a formula that was used in step three would not of worked for this step, as the hidden faces results did not match that of step two and three combined.

Here are my future predictions on a chart:

Forth Set:

Total Number Of Faces | Hidden Faces | Faces In View |

168 | 118 | 50 |

192 | 136 | 56 |

216 | 154 | 62 |

240 | 172 | 68 |

264 | 190 | 74 |

Compared to Second set and third set combined:

Here are my future predictions on a chart:

Total Number Of Faces | Hidden Faces | Faces In View |

144 | 82 | 62 |

168 | 97 | 71 |

192 | 112 | 80 |

216 | 127 | 89 |

240 | 142 | 98 |

Through observing this chart, I can see that there is a match in the total number of faces after 168, although the hidden faces and faces in view are completely different. The difference in to charts would be because the second and third set is not actually combined to form one shape, they are just two different results added together. The forth set is actually a shape, this explains why there is a match with the total number of hidden faces. This set has once again shown hat the shape itself affects the number of hidden faces and the way it is positioned, not the number of total faces.

My next set of cuboids will be aimed at proving something I have covered in the previous sets, that the more faces facing the base equals more hidden faces. I also to find a formula that doesn’t rely on adding a certain amount of cubes to the end of it for the hidden face formula.

Fifth set of cuboids

15,25,35,45

27,89,211,321

62,122,110

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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