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Problem of the Week
Problem A and Solution
3-d Fun

Problem

The three-dimensional figure shown was built using interlocking cubes.

The front, the top, and the right side of the figure are visible. The figure has three layers: top, middle and bottom. Each layer has a front row, a centre row and a back row. In the bottom layer, four cubes are visible. These cubes are front left, front right, centre right, and back right. In the middle layer, three cubes are visible. These cubes are front right, centre left and centre right. In the top layer, two cubes are visible. These cubes are centre left and centre right.

When we look at the figure directly from the top, we will see the following image, which is called the top view.

Five identical squares form a shape. The shape looks like a rectangle that is three squares high and two squares wide but is missing the top left square.

  1. How many cubes in total are in the three-dimensional figure? Is there more than one possible answer?

  2. Draw the front view and side view of the figure.

Solution

  1. To count the total number of cubes, we will look at the three-dimensional figure one layer at a time. It has a top layer, a middle layer, and a bottom layer.

    First we will look at the top layer. There are \(2\) cubes in this layer, as shown.

    Two cubes placed side by side.

    Next we will look at the middle layer. We can see \(3\) cubes in this layer, however we cannot see from the three-dimensional figure whether or not there is a cube behind the striped cube shown.

    The centre left cube in the middle layer of the figure is striped.

    However the top view shows that there is no cube in this position, so we can confirm there are only the \(3\) cubes shown in the middle layer.

    Three cubes placed side by side forming a capital L shape.

    Finally, we will look at the bottom layer. We can see \(4\) cubes in this layer. However, we cannot see from the three-dimensional figure whether or not there is a cube directly under the striped cube. The top view does not give us any more information here, because the position in question would not be seen from the top view as it would be blocked by the cube above the striped cube. Thus, without more information, we must conclude that this layer contains either \(4\) or \(5\) cubes, as shown.

    Five identical cubes are arranged in a layer. The shape looks like a rectangular prism that is two cubes wide and three cubes deep but is missing the back left cube. OR   Four identical cubes are arranged in a layer. The shape looks like a rectangular prism that is two cubes wide and three cubes deep but is missing the back two cubes on the left side.

    Thus, in total, the three-dimensional figure has \(2+3+4=9\) or \(2+3+5=10\) cubes.

    Note that if the figure had been made using regular stacking cubes instead of interlocking cubes, it would not be possible to have a gap in the bottom layer, so the bottom layer would have to contain \(5\) cubes.

  2. The front view is what we see when we look at the three-dimensional figure directly from the front. We would see the image below.

    Six identical squares are arranged to form a rectangle that is three squares high and two squares wide.

    The side view is what we see when we look at the three-dimensional figure directly from the side. We would see one of the images below, depending on which side of the figure we look at.

    Six identical squares form a shape. The shape looks like a larger three by three square with the top left, top right and middle right squares removed. OR Six identical squares form a shape. The shape looks like a larger three by three square with the top left, middle left, and top right squares removed.

Teacher’s Notes

Students often enjoy studying math in elementary and high school, but 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 \(2\)-dimensional models from a \(3\)-dimensional object. 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.