The Painted Cube Problem

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The Painted Cube Problem
By Becky and Lydia 80 The IN.BB.LG team
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Becky Barnes And Lydia Gill
Names Becky and Lydia . Age Both 12 years
old. School Settlebeck High School, Sedburgh,
Cumbria, England Project Painted Cube Problem
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Isaac Newton (1642-1727)

• Isaac Newton was an English mathematician, who
  discovered the binomial theorem, (A theory he
  came up with) he also invented calculus, and
  produced theories of mechanics, optics, and the
  law of universal gravitation. Many of his ideas
  for which he is famous were developed in
  isolation in the year 1665 during the Great
  Plague. He was also knighted Sir Isaac Newton.

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You Will Need

• For this slideshow you will need
• Isometric paper,
• Pencil,
• Four different coloured pencil crayons,
• Calculator,
• AND YOUR BRAIN!

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Easy Problems
This cube has 6 faces painted.
For our first problem we are going to show you is
the 1 by 1 by 1 cube. So to build the 1 by 1 by
1cube all you need is 1 cube. Then all you need
to do is to count the number of cubes which have
all of its faces panted.
All the cubes on this cube, all have 3 faces
painted.
For our Second we are going to show you is the 2
by 2 by 2 cube. So to build the 2 by 2 by 2 cube
all you need to do is get 8 cubes and put them
together like the diagram above. Then all you
need to do is count how many cubes have three
faces painted.
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A Table To Keep your results in
Cubes with 0 faces Painted Cubes with 1 Face Painted Cubes with 2 Faces Painted Cubes with 3 Faces Painted Cubes with 6 Faces Painted Total number of cubes
1 by 1 by 1 0 0 0 0 1 1
2 by 2 by 2 0 0 0 8 0 8
3 by 3 by 3 Answers will be shown Later on in the slideshow
4 by 4 by 4
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How to Count The Number Of Faces Painted On A
Cube

• Counting the faces on each individual cube sounds
  easy but its not. If you wont it to be easy
  follow these simple steps.
• Draw your cube on isometric paper.
• Colour all the cubes which all have three faces
  painted blue.
• Colour all the cubes which all have two faces
  painted green.
• Colour all the cubes which all have one face
  painted brown.
• Remember if on the inside of the cube there are
  cubes with no faces painted to count them ones.
• Remember you can not see one of the cubes which
  has three faces painted because you can not draw
  it on a piece of isometric paper.
• Colouring in represents painting the the cube.

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Key
3 faces painted
2 Faces painted
1 face painted
0 Faces painted
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Easy painted cube problems
In a 3 x 3 x 3 cube there are1 cubes with 0
faces painted.6 cubes with 1 face painted.12
cubes with 2 faces painted.8 cubes with 3 faces
painted.
Here is a 3x3x3 cube.
In a 4 x 4 x 4 cube there are8 cubes with 0
faces painted.24 cubes with 1 face painted.24
cubes with 2 faces painted.8 cubes with 3 faces
painted.
Here is a 4x4x4 cube
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A Table To Keep your results in
Cubes with 0 faces Painted Cubes with 1 Face Painted Cubes with 2 Faces Painted Cubes with 3 Faces Painted Cubes with 6 Faces Painted Total number of cubes
1 by 1 by 1 0 0 0 0 1 1
2 by 2 by 2 0 0 0 8 0 8
3 by 3 by 3 1 6 12 8 0 27
4 by 4 by 4 8 24 24 8 0 64
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An exploded cube
An exploded cube is a really simple way of
working out the painted cube problem. First you
explode a cube like the diagram at the bottom.
Then you colour all the sides if they are 3 faces
painted sides or 2 faces painted sides or if they
are 1 face painted side or if they are 0 faces
painted sides. This will help you by making the
painted cube problem a lot easier. You try and
draw an exploded cube on your isometric paper.
This is the centre cube which does not get
painted but we have just highlighted it to make
it easier to see
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Information on a exploded cube
All these cubes have one of there faces painted.
These are all cubes which have two of there faces
painted.
This cube has three of its faces painted.
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Now you try
Work out how many cubes have 3 faces painted 2
faces painted 1 face painted 0 faces painted
Here is a 5 by 5 by 5 cube
Here is a 6 by 6 by 6 cube
Work out how many cubes have 3 faces painted 2
faces painted 1 face painted 0 faces panted
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Answers
The answer to the 5 by 5 by 5 panted cube is 8
cubes have 3 faces painted 36 cubes have 2 faces
painted 54 cubes have 1 face painted 27 cube has
0 faces painted
The answer to the 6 by 6 by 6 8 cubes have 3
faces painted. 48 cubes have 2 faces painted. 96
cubes have 1 face painted. 64 cubes have 0 faces
painted.
Were you right!
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Try an exploded cube yourself
How many cubes have 3 faces painted 2 faces
painted 1 face painted 0 faces painted Total
number of cubes 27
This is an 3 by 3 by 3 exploded cube.
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Try a 4 by 4 by4 by 4 exploded cube.
How many cubes have 3 faces painted 2 faces
painted 1 face painted 0 faces painted Total
number of cubes
This is a 4 by 4 by 4 exploded cube.
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Answers
The answers to the last two exploded cube
problems.
The answers to the 4x4x4 cube are
The answers to the 3x3x3 cube are
In a 4 x 4 x 4 cube there are8 cubes with 0
faces painted.24 cubes with 1 face painted.24
cubes with 2 faces painted.8 cubes with 3 faces
painted.
In a 3 x 3 x 3 cube there are1 cubes with 0
faces painted.6 cubes with 1 face painted.12
cubes with 2 faces painted.8 cubes with 3 faces
painted.
Hope you got it right!!!!!
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How to work out the total of cubes faster!
To work out the total number of cubes in a cube
lets say it was a 10 by 10 by 10 cube. You would
do 10 x 10 x 10 that answer would be how many
cubes are in a10 by10 by 10 cube.
To work out the total number of cubes that have 3
faces painted in any cube is very easy because
all cubes have 8 cubes with 3 faces painted, if
the cube dose not have 8 cubes with 3 faces
painted it is not a cube.
To work out the total number of cubes have 2
faces painted in any cube, you find out how many
cubes are along the edge of one side then
takeaway 2 then times that number by 12 because
there are 12 edges on a cube.
To work out the total number of cubes which have
1 face painted you, work out how many cubes are
on a face by doing 10-2 then you square that
number. Then you times that number by 6 because
there are 6 faces on a cube.
To work out the total number of cubes with 0
faces painted, you takeaway 2 from the 10. Then
you do 888.
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The End!
We both hope you have enjoyed our side show and
hope it has taught you some things about the
painted cube problem!
By Becky and Lydia!
Goodbye and thank you for joining us!
Click to go back to the start