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Problem of the Week
Problem A and Solution
3-D Fun
Problem
The three-dimensional figure below was built using interlocking cubes.
Assume this is the top view of the 3D figure
How many cubes are in the figure?
Draw the other views (front, right, left, back).
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Solution
There are either 9 or 10 cubes in the figure.
Looking at this 3D figure from the front you would see a grid
that is 3 blocks high and 2 blocks wide. You do not notice
that there is a single block on the left side of the front row,
or that there are only 2 stacked blocks on the right side of
the front row.
Looking at this figure from the right you would see all of the
blocks on the right side. This includes the 2 blocks at the
right, front, the 3 blocks at the right, middle, and the single
block at the right, back of the figure.
The left view is like the right view, but the mirror reflection.
Based on the top view, you know that there are no blocks
in the back left corner of the figure. Based on the top-right
view, you know that there are no blocks on the left side
that are higher than the right side in the front or middle
of the figure. However, since these are interlocking blocks,
we could attach them to each other on any side. In other
words, you are not required to stack blocks on top of another
block. This means you do not have enough information to
determine if there is a block at the bottom, left, middle
location. This does not affect the view, but it does affect
the possible total number of blocks in the figure. If the
figure is made from regular stacking blocks, it is not possible
to have a gap at the bottom. Using regular blocks, the figure
must have 10 cubes.
The back view would appear the same as the front view.
The blocks in the middle of the figure fill the grid using this
view.
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Teacher’s Notes
Some students like studying math in elementary and high school, but they do not
know what they could do at the next level. A degree in mathematics can lead to
many different careers. Many of the top jobs in lists ranking the best occupations
require analytical math skills. Math shows up in some unexpected places.
The CEMC has created a resource called Real-World Math
(http://cemc.uwaterloo.ca/resources/real-world.html) that gives insight
into how math is used in areas such as security, medicine, and the environment.
This problem has students visualizing a 3-dimensional object from a
2-dimensional model. Architects, engineers, and computer animators need to
create models for buildings, tools, and images. These occupations, and many
others rely heavily on a solid understanding of mathematics.