Cube Towers Investigation

Aim The aim of this investigation was to find how many hidden faces there were when various sized cuboids were built, and also to find the relationship between the number of hidden faces in a cube and how many cubes there were in the shape.

Square Based Towers I started off the investigation using square based cuboids, eg. 1 x 1, 2 x 2, 3 x 3, etc. 1.1 Diagrams

n=1 x=1

n=2 x=3

n=3 x=5

n=4 x=7

n=5 x=9

n=6 x = 11

n=7 x = 13

1.2 Table of Results 1 x 1 CUBES No. of Cubes (n) 1 2 3 4 5 6 7 1.31 Pattern (WORDS)

Height (h)

Hidden Faces (x)

1 2 3 4 5 6 7

1 3 5 7 9 11 13

The number of hidden faces goes up by two each time another cube is added

on. 1.32 Pattern (ALGEBRA) The formula to find the number of hidden faces in a 1 x 1 square-based cube tower is 2n – 1. The 2n comes from the pattern that each time another cube is added to the previous one(s), the number of hidden faces go up by two; but that can’t be the only part of the formula, because then the linear sequence would be 2, 4, 6, 8, 10… etc. To get from that to 1, 3, 5, 7, 9… etc, you would have to minus one, hence the formula 2n – 1. 1.4 Checking My Formula EXAMPLE #1 = 2(3) – 1 EXAMPLE #1 = 6 – 1 EXAMPLE #1 = 5 EXAMPLE #2 = 2(5) – 1 EXAMPLE #2 = 10 – 1 EXAMPLE #2 = 9 1.5 Prediction EIGHT CUBES = 2(8) – 1 EIGHT CUBES = 16 – 1 EIGHT CUBES = 15 1.6 Checking My Prediction

2 hidden faces 2 hidden faces 2 hidden faces 2 hidden faces 2 hidden faces 2 hidden faces 2 hidden faces 1 hidden face

1.7 Explanation

15 hidden faces My prediction was correct, which means my formula is too.

The formula, 2n – 1, works because every time another cube is added to the tower, two new hidden faces are created, which is where the 2n comes from.

every new cube creates two hidden faces BUT … the first cube only has one hidden face A [ – 1 ] part has to be included in the formula as well. This is because the first cube in the tower only has one hidden face when it’s by itself; if the formula is just 2n, it will imply that EVERY cube has 2 hidden faces, which isn’t true. This is why the formula 2n – 1 works.

2.1 Diagrams

n=4 x = 12

n=8 x = 28

n = 12 x = 44

n = 16 x = 60

n = 20 x = 76

n = 24 x = 92

n = 28 x = 108

2.2 Table of Results 2 x 2 CUBES No. of Cubes (n)

Height (h)

Hidden Faces (x)

4 8 12 16 20 24 28

1 2 3 4 5 6 7

12 28 44 60 76 92 108

2.31 Pattern (WORDS) The number of hidden faces goes up by sixteen each time another row consisting of four cubes is added on. 2.32 Pattern (ALGEBRA) The formula to find the number of hidden faces in a 2 x 2 square-based cube tower is 4n – 4. The 4n comes from the pattern that each time another four cubes are added to the previous row, the number of hidden faces goes up by sixteen. The [ – 4 ] is taken from how sixteen relates to the number of hidden faces. When there are four cubes, the number of hidden faces is twelve, but we know the hidden faces increase by sixteen each time, so to get from sixteen to twelve, we have to subtract four. When there are eight cubes, the number of hidden faces is twenty-eight, which is 4(8) – 4, which is how I figured out the formula to be 4n – 4. 2.4 Checking My Formula EXAMPLE #1 = 4(12) – 4 EXAMPLE #1 = 48 – 4 EXAMPLE #1 = 44 EXAMPLE #2 = 4(20) – 4 EXAMPLE #2 = 80 – 4 EXAMPLE #2 = 76 2.5 Prediction EIGHT CUBES = 4(32) – 4 EIGHT CUBES = 128 – 4 EIGHT CUBES = 124 2.6 Checking My Prediction

n=8 x = 124

3.1 Diagrams

n=9 x = 33

n = 18 x = 75

n = 27 x = 117

3.2 Table of Results 3 x 3 CUBES No. of Cubes (n)

Height (h)

Hidden Faces (x)

9 18 27 36 45 54 63

1 2 3 4 5 6 7

33 75 117 159 201 243 285

3.31 Pattern (WORDS) The number of hidden faces goes up by forty-two each time another row consisting of nine cubes is added on. 3.32 Pattern (ALGEBRA) The formula to find the number of hidden faces in a 3 x 3 square-based cube tower is 42/3n – 9. The 42/3n comes from the pattern that each time another nine cubes are added to the previous row, the number of hidden faces goes up by forty-two. I calculated that for nine cubes, a value multiplied by nine must equal forty-two, which, in turn, when subtracted by nine would equal thirty-three; for eighteen cubes, that same value multiplied by eighteen would equal eighty-four, and when subtracted by nine, would equal seventy-five. The [ – 9 ] is taken from how forty-two relates to the number of hidden faces. When there are nine cubes, 42/3n part of the formula would give me forty-two, and in order to get from that to thirty-three, I subtracted nine from it, which gave me the formula 42/3n – 9. 3.4 Checking My Formula

EXAMPLE #1 = 42/3(27) – 9 EXAMPLE #1 = 126 – 9 EXAMPLE #1 = 117 EXAMPLE #2 = 42/3(45) – 9 EXAMPLE #2 = 210 – 9 EXAMPLE #2 = 201 3.5 Prediction EIGHT CUBES = 42/3(72) – 9 EIGHT CUBES = 336 – 9 EIGHT CUBES = 327 3.6 Checking My Prediction

n = 72 x = 327

4.1 Diagrams

n = 16 x = 64

n = 32 x = 144

n = 48 x = 224

4.4 Table of Results 4 x 4 CUBES No. of Cubes (n)

Height (h)

Hidden Faces (x)

16 32 48 64 80 96 112

1 2 3 4 5 6 7

64 144 224 304 384 464 544

4.31 Pattern (WORDS)

The number of hidden faces goes up by eighty each time another row consisting of sixteen cubes is added on. 4.32 Pattern (ALGEBRA) The formula to find the number of hidden faces in a 4 x 4 square-based cube tower is 5n – 16. Each time another row is added, the number of hidden faces increases by eighty. The 5n shows the relationship between eighty and the total number of cubes. The [ – 16 ] comes from eighty subtract sixteen equals sixty-four; one hundred and sixty subtract sixteen equals one hundred and fourteen; etc. 4.4 Checking My Formula EXAMPLE #1 = 5(48) – 16 EXAMPLE #1 = 240 – 16 EXAMPLE #1 = 224 EXAMPLE #2 = 5(80) – 16 EXAMPLE #2 = 400 – 16 EXAMPLE #2 = 384 4.5 Prediction EIGHT CUBES = 5(128) – 16 EIGHT CUBES = 640 – 16 EIGHT CUBES = 624 4.6 Checking My Prediction

n = 128 x = 624

2.7 ; 3.7 ; 4.7 Explaining the Formulae To find the formula for the different square-based cuboids, all I did was follow a few simple steps (example used from 2 x 2 cube tower). 1. Figure out how much the hidden faces go up by each time, eg, 12  28, 28  44, etc. would be 16.

12 faces  28 faces

2. Calculate (that number) ÷ (the number of cubes in the first row), which gives you the _n number, eg, 16 ÷ 4 = 4, so the first part of the formula will be 4n. 3. To get the second part of the formula, use the number by which the hidden faces go up by each time again, and subtract the number of hidden faces if the height is one from it, eg, 16 – 12 = 4. So – 4 would be the second part of the formula.

The General Formula After finding the formula for finding the number of hidden faces in each square-based tower, I tried to find a general formula that not only would be able to solve the hidden faces in all square-based towers, but also to be able to find hidden faces in rectangular-based towers as well. I realized that what I was really looking for was just a formula that showed: total faces – seen faces = hidden faces The total faces could be seen as 6n. The seen faces can also be referred to as the area.

h

l

There are two (length by height) sides, so that would be2lh.

w

There are two (width by height) sides, so that would be2wh.

There is only one (length by width) side, because the bottom can’t be seen, so it would be lw.

General Formula for Finding Hidden Faces in All Cuboids

6n – (2lh + 2wh + lw) Conclusion In conclusion, the formula for finding the number of hidden faces in all cuboids is 6n – ( 2lh + 2wh + lw ).