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SED ME 501
Summer 2010
Cube Problems
1. Sixty-four unit cubes were glued together to make a 4x4x4 cube. How many pairs of faces,
one-by-one unit each, were glued together to make the larger cube?
2.
A cube is painted red and then cut into 64 identical smaller cubes. How many of
these cubes are painted red on at least one face?
3.
A cube is painted red and then cut into 1000 identical smaller cubes. How many of
these cubes are painted red on at least two faces?
4.
A cube is made by gluing together 1000 identical smaller cubes. The cube is painted
blue, and then taken apart. What percent of the small cubes are unpainted?
5.
A large cube is made by gluing together identical smaller cubes. The cube is painted
blue, and then taken apart. The number of unpainted cubes is the cube of the number
of cubes painted on three sides. That is, (number with three sides painted)3 = (number
of unpainted). How many small cubes were used to make the large cube?
6.
Cubes in Space: Imagine a 10 by 10 by 10 cube made up of 1000 unit cubes and
floating in space. What is the greatest number of unit cubes that can be seen by an
observer at any moment in time?
7.
A cube with edges measuring 10 cm is dipped into red paint. The cube is then
divided into 125 smaller cubes. One cube is drawn, at random. What is the
probability that the selected cube will have at least 25% of its surface area painted
red?
8.
A 5x5x5 box without a lid is completely filled with 125 cubes. How many of the
cubes touch a side or the bottom of the box?
9.
A wooden cube is painted red and then cut with six cuts into equal cubes. If the
cubes are placed in a bag and one cube is drawn from the bag, what is the probability
that it will have at most one side painted red?
10. Unit cubes are glued together to make a cube several units on each side. Some of the
faces of this large cube are painted. When the cube is taken apart, there are exactly 45
cubes without any paint. How many faces of the large cube were painted?
11. The Mathematicians Banquet. At the Mathematicians’ Banquet, there was a cubeshaped cake that was frosted on all sides, and cut into smaller cube-shaped pieces.
Use the geometry of the cube and the algebra of patterns to solve the question: How
many mathematicians were there at the banquet? You know that there were just
enough pieces for everyone, and there were 8 times as many pieces with frosting on 0
sides as there were with frosting on 3 sides.
Boston University
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Carol Findell
SED ME 501
Summer 2010
12. Mariko has a set of cubic blocks. She can arrange all her blocks to make a cube
(with more than one block) or to make a square. What is the least number of blocks
she can have?
13. Leland also has a set of cubic blocks. He can arrange all his blocks to make three
different sized squares in two different ways. He can also arrange them to make two
different sized squares. How many blocks does he have?
(Answer: 74: 74 = 9 + 16 + 49; 74 = 1 + 9 + 64; 74 = 49 + 25)
14. Twenty unit cubes are glued together to form this
3 x 3 x 3 figure, with “holes” that you can see
through. The figure is dropped into a bucket of
red paint. After it is taken out and dried, how
many square units of surface area are painted red?
More Cube Problems
1. A rectangular box is 2 cm high. 4 cm wide, and 6 cm deep. Michelle packs the
box cubes, each 2 cm by 2 cm by 2cm, with no space left over. How many cubes
does she fit into the box?
2. The tower shown at the right is made
by placing congruent cubes on top of
each other with no gaps. Not all cubes
are visible. How many cubes does the
tower contain?
3.
4. The stairway at the right is made by placing identical
cubes on top of each other. Not all cubes are visible.
How many cubes does this stairway contain?
5. A large cube, 5 cm by 5 cm by 5 cm, is painted orange on all six faces. Then it is
cut into 125 small cubes, each 1 cm by 1 cm by 1 cm. How many of the small
cubes are not painted orange on any face?
Boston University
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Carol Findell
SED ME 501
Summer 2010
6. A supermarket clerk makes a solid pyramid out of
identical cereal boxes. The top five layers are
shown. What is the total number of cereal boxes
in these top five layers?
7. A cube has 6 faces: top, bottom, and all 4 sides. The
object is made of six congruent cubes. Not all faces
are visible. All outer faces of the object including the
bottom are painted blue. How many faces of the
cubes are painted blue?
8. Mark has 42 identical cubes, each with 1-cm edges. He glues them together to
form a rectangular solid. If the perimeter of the base is 18 centimeters, find the
height of the rectangular solid, in cm.
8. Using this staircase of 1-cm cubes, how many more
1-cm cubes will you need to make a cube
measuring 5 cm on each side?
9. A cubical box without a top is 4 cm on each
edge. It contains 64 identical 1 cm cubes that
exactly fill the box. How many of these small cubes actually touch the box.
10. The 4 x 4 x 4 cube was made from 64 white cubes.
Then black paint was painted on some squares on
each face. Opposite faces were painted the same
way. How many of the 64 cubes have black paint on
at least one face
Boston University
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Carol Findell