and its magicNatasa Dragovic

University of Texas at Austin

Cube or hexahedron

Cube – that is a figure we see not only in the geometry and art classes, but also in every day life. The other

name for cube is a regular hexahedron.

Cube is a regular polyhedron with square as its face. We can also call cube 3D square. Cube has 8 vertices, 6 faces and 12 edges. Cube is an amazing geometric figure, into which one can inscribe other figures, such as octahedron, tetrahedron, icosahedron, etc.

Platonic solids

Cube always had a lot of puzzles in it – it’s a

surprisingly complex and at the same time surprisingly simple geometric object, which helps look into the depth of consciousness.

Even in ancient times Plato called it a sacred figure and related it to the symbol of Earth, because it is the most stable of all the

figures.

Optical illusions - The Necker cube

Is the red dot on the near or far corner?

Impossible objectThe impossible cube or irrational cube is an impossible

object invented by M.C. Escher for Belvedere, a lithograph in which a boy seated at the foot of the building holds an impossible cube.

Unusual houses the ordinary world

Cube houses. Rotterdam. Netherlands

Kubuswoningen – cube house – project of the architect Piet Blom.

Unusual houses the ordinary worldLaTeteauCarre in Nice, France.

This sculpture is called LaTeteauCarre or translated to English – thinking inside the box. The building was made by the sculptor and artist Sacha

Sosno. Project was made for the central library in Nice. The height of the building is 26m, and inside the head are the 3 floors of the library, and most

come just at the ‘cube’, which is divided into tiers within.

Unusual houses the ordinary worldCube Tube

Chinese city Jinhua became famous in 2010 for having built a complex made of two cube shaped buildings.

Puzzle toys

The most famous toy puzzle, that every child has held at least once

and some even managed to assemble is Rubik’s cube. As the

name says, this world famous puzzle toy was created by

Hungarian architect Erno Ribuk in 1975. Rubik’s cube is a plastic cube

made of 26 small cube. Once Rubik’s cube is assembled, each

face has one color.

Rubik’s cube

Puzzle toys

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube. The pieces can also be used to make a variety of other 3D shapes.

Soma cube

FACE

EDGE VERTEX

Vertices –

Edges –

Faces –

12

8

6

Cube

Opposite facesProblem 1: A cube has six different symbols drawn over its six faces. The symbols are dot, circle, triangle, square, cross and arrow. Three different positions of the cube are shown in figures X, Y, and Z.

(X) (Y) (Z)

Which symbol is opposite the dot?

A. Circle B. Triangle C. Arrow D. Cross

Opposite faces

Problem 2: Here are three views of the same cube. Each face is painted a different color.

What is the color of the face opposite to the yellow one?

Answer: blue.

Opposite facesProblem 3: Each face of a cube is colored in one of the colors: white, pink, green, yellow, blue and red. Viewing from one side, one sees blue, yellow and white faces. From another point, one sees pink, blue, and red faces. From a third point, one sees green, pink, and white faces. What is the color of the face opposite to the white one?

y

w

g r, p

Answer: The red face is opposite to the white one.

?

Opposite facesProblem 4: On cube faces are written numbers from 1 to 6. Cube is thrown twice. First time, the sum of numbers from the four side faces was 12, second time - 15. What number is written opposite of face that has number 3 written on it?

Solution: Let’s notice, that the total sum of numbers written on the cube is 21. The sum of numbers of top and bottom face in the first and second case is 9 and 6 respectively.

Answer:  6.

After the first throw, it is clear that it’s either 3 across from 6, or 4 across from 5. Let’s assume it is 4 across 5. But after the second throw, it is clear that it’s either 1 across from 5, or 2 across from 4. Which is a contradiction. Therefore 3 is across from 6.

A net can be transformed into a cube.

How many different nets can you draw that can fold into a cube?

A cube has eleven nets.

Nets of a cubeProblem 5: Choose the box that is similar to the box formed from the given sheet of paper (X).

(X)

(1) (2)

(3) (4)

Answer: (4).

Nets of a cubeProblem 6: On each face of a paper cube, number 1, 2 or 3 is written. On the opposite faces, numbers are equal. Which of the following figures is possible to obtain if we cut the cube along few edges and unfold it?

3 1 3 2

2

1

1 3 2 3

1

2

3

1231

2

1232

1

3

2121

3

3

(A) (B) (C)

(D) (E)

Answer: (E)

Nets of a cubeProblem 7: Cube is cut with a plane. On the net line traces of the cut are shown. What shape did the cut make with the cube?

(A) Equilateral triangle

(B) Rectangle, but not square

(C) Right triangle

(D) Square

(E) Hexagon

24

1

5 6

3

Answer: (A) equilateral triangle

Nets of a cubeProblem 8: For two cubes, three nets were made and they were mixed together (fig. A-F). Find nets of each of the cube’s.

(A) (B) (C)

(D) (E)(F)

Answer: (A, C, F); (B, D, E)

Nets of a cubeWe can also pose the problem other way around. Here you have a single 3-dimensional cube and a number of 2-dimensional patterns, only one of which when folded, will make the cube.

Problem 9: Which of the patterns when folded will make the cube shown?

A B C D

Answer: (C)

Nets of a cubeProblem 10: From paper cube an edge was cut of. This cube was then cut along few edges and we got one of the following figures A - E. Which one?

(A) (B) (C)

(D) (E)

Answer: (E)

Nets of a cubeProblem 11. A paper strip has 3 cm width. What is the length of the shortest strip, from which one can assemble a cube of edge 3 cm? The assembled cube has to have all six faces.

Solution: The shortest paper strip of width 3 cm, from which we can assemble a cube of dimensions 3 х 3 cm, has length 21 cm. How to assemble this strip is shown in the picture:

If we color the strip on one side in black, then in order to assemble a cube with all six black faces, we would need a stripe of length 24 cm.

Nets of a cubeProblem 12: We are given a cube with edge length 1. One of its faces was glued to the central square of 3х3 square. Explain, how to wrap this cube into the paper sheet if you can only cut along the grid and bend it.

Solution: The key is to make the cube net from the 3x3 square.

1. We plan the cut.

2. We compose it.

Painting and cuttingProblem 13: Imagine a wooden cube with 3 cm edge, each of which is colored red.

1. How many cuts are needed to break the cube into small cubes with 1 cm edge?

2. How many small cubes do we get?

3. How many cubes will have 4 colored faces?

4. How many cubes will have 3 colored faces?

5. How many cubes will have 2 faces colored?

6. How many cubes will have 1 colored face?

7. How many uncolored cubes are there?

6

27

0

12

8

6

1

Painting and cuttingThe number of small cubes of edge “x” unit can be cut from the large cube of edge “A” unit is (A/x)3.

The following rules must be followed when a big cube in painted. If A unit be the edge of the large cube and x unit is the edge of the small cube, then

a) The number of small cubes painted from 3 adjacent faces is always 8.

b) The number of small cubes painted from 2 adjacent faces 12 (A/x – 2).

c) Number of small cubes painted from one faces only 6 (A/x – 2)2 .

Painting and cuttingProblem 14: A cube of side 4 cm is painted with 3 colors red, blue and green in such a way that opposite sides are painted in the same color. This cube is now cut into 64 cubes of equal size.

1. How many have at least two sides painted in different colors? 24 + 8 = 32

2. How many cubes have only one side painted? 24

3. How many cubes have no side painted? 8

4. How many have exactly one side not painted?0

Painting and cuttingProblem 15: We have a cube with edge length 2. Show how to glue on it 10 squares with edge length 1 so that no two squares touch (do not share edge or part of it). You cannot bend the squares.

Solution: One of the possible examples is shown in the picture. For convenience, squares are shown on the net of the cube.

Painting and cuttingProblem 16: Big cube is glued from small wooden cubes. It was cut through on 6 places, all parallel to the edges (fig.). How many small cubes are left unharmed?

Answer: 44 small cubes.

Painting and cuttingProblem 17: Termite and 27 cubes.  Lets assume we have a big cube, which is composed from 27 small wooden cubes (fig.).

Termite sits on the middle face of one of the outer cubes and cuts through a path that goes through all the cubes. After going through one cube, termite does not return. He is always moving parallel to some edge of the big cube, but never parallel to the diagonal.

Can the termite cut through all 26 inner cubes, go through each once and finish his journey in the central cube? If it is possible, show what should be termite’s path. If you think it is impossible, prove it.

Painting and cuttingSolution: Let’s color the cubes like a chess board. Meaning that each two neighboring cubes should have different color: one white, the other one – black. Then, moving by the conditions of the problem, termite has to go from one color cube to the other one. Changing cube colors is an invariant property of this problem. Let’s say termite started moving from the black cube.

Lets make a table of change of color during termite move.

Number’s cube 1 2 3 4 . . . 26 27

Color of a cube B W B W W B

Meaning that the last 27th cube has to be black. However from the image it is clear that the central cube has to be white. Therefore we got to the contradiction,

which tells us that termite cannot make the path described in the problem.

Problem 18: In Guggenheim museum in New York City there is a sculpture that has cube shape. Cockroach sitting on one of the vertices, wants to pass as fast as possible around the sculpture so that he can move on to the other exhibits (for that it is enough to get to the opposite vertex of the cube). What path should the cockroach choose? 

Find the shortest trajectory

Find the length of the pathProblem 19: Find the shortest path length along the surface of unit the cube ABCDA1B1C1D1, which connects vertices A and C1.

1

1

1

A1 B1

C1D1

D

B

C

A

A1B1 C1

E

CB

A11

1

Solution: Lets look at the net that is made from two adjacent faces of the cube.

The shortest path from A to C1 is the segment AC1, length of which is equal to .

The correspondent path is drawn on the surface of the cube.5

E

Lets notice that the path from A to C1 is not unique. There are six such paths of

length , that go through middle of edges BB1, A1B1, A1D1, DD1, CD и BC.

Answer: 5

5

Find the length of trajectoryProblem 20: On the surface of the Rubik’s cube, can you draw a closed path, such that it goes through each square exactly one time (path does not go through the vertices of the squares)?

Hint: On each face we can draw a face that goes through all 9 squares of that face, and at the same time it starts at any given corner and ends at any other corner.

Solution: We show one example of such a path.

Answer: You can.

Computational problemsProblem 21: Number the 8 vertices of the cube with numbers (1, 2, 3, 4, 5, 6, 7, 8), so that the sum of each six of its faces were equal.

4 7

61

8 3

25

Solution: Each vertex belongs to three faces, so we should take the sum 1 + 2 + ... + 8 and multiply it by 3, and then divide by 6 ( number of faces), and we get 18 – sum of numbers of each face.

Computational problemsProblem 22: Can you number each edge, such that sum of number on edges that share a vertex are equal if the numbers are:

а) 1, 2, ... , 12; b) -6, -5, ... , -1, 1, 2, ... , 6?

Answer: а) No. Lets assume, that it is possible, and that the sum of edges sharing a vertex is equal to х. Then the sum on all 8 vertices is equal to 8х. On the other hand, since each number entered to this sum twice, this sum is also equal to:

(1 + 2 + ... + 11 + 12) 2 = (1 + 12) 12 = 156. Equation 8х = 156 does not have an integer solution, so our solution is wrong.

b) Yes. The sum of edges that share a vertex is equal to 0.

-1

-3

-5

-4

-6-2

1

5

3

62

4

Figures from cubesProblem 23: Put together a cube 3×3×3 from the red, yellow and green cubes 1×1×1 so that in each segment of size 3×1×1 there are cubes of all 3 colors.

Answer: Color by layers:

Figures from cubesProblem 24: We are given 27 cubes of equal size: 9 red, 9 blue and 9 white ones. From these cubes can we put together a cube so that each column of three cubes contains exactly two colors? (We look at columns that are parallel to the edges of the cube, so we look at 27 columns)

Solution: In the picture we show layers of the cube 3×3×3, that are composed out of 27 color cubes.