Batch 2 - Class 270 - Spatial Visualization - Cubes and Perspectives

• A 5 by 5 by 5 cube is formed using identical 1 by 1 by 1 cubes. A number of the smaller cubes are removed by punching out the fifteen designated columns from front to back, top to bottom, and side to side. What percentage of the volume of the original cube remains following the removal of the fifteen columns?

• Answer 54.4% (Only 68 of surface cubes remain - inside is hollowed)

• Suppose all six faces of an n × n × n are painted red. Then one of the n^3 unit cubes is randomly selected and tossed like a die. What is the probability that the face obtained is painted? 
• Answer: 1/n. There are 6n^2 painted faces out of total of 6n^3 (the problem can also be solved by taking all cases of cubes with different number of painted sides)
• Bob and Ann play the following game with 8 white unit cubes. Ann wins if she can assemble a 2 × 2 × 2 cube that has only white faces exposed. But Bob gets to paint four of the 8 · 6 = 48 white faces black. Who wins? 
• Answer: Bob - if he paints 4 sides of a single cube, then one can't assemble them into a white 2x2x2 cube
• How about a 3x3x3 cube - how many faces need to be painted?
• Answer 12 - the middle one, and then one more with all sides
•  Three cubes with volumes 1, 8, and 27 are glued together to obtain a solid polyhedron with minimal surface area. What is that surface area?
• Answer: The combined surfaces of the three cubes is 6(9 + 4 + 1) = 6·14 = 84. We can reduce the surface area of the polyhedron by 2·4 = 8 when we put the two larger cubes together. Then we can reduce the surface area of the polygon by 4 more by gluing the unit cube so that it has a face in common with each of the other two cubes. So the minimal surface area of the polyhedron is 84 − 8 − 4 = 72 square units. 
• What will the following shapes fold up to?

• Answer: Prism and pyramid (tetrahedron)

• Explain the notion of perspectives to kids, by taking a 3D object and drawing its three perspectives.
•  Find all possible cubical buildings that have a base, and front projection and a right side projection that is as follows

• Answer: 7 solutions. 2 with 6 cubes, 4 with 7 cubes, 1 with 8 cubes
• Is there number of cubes determined by the following perspectives

• Answer: Yes

• Is there number of cubes determined by the following perspectives

• Answer: Yes

• Use exactly 20 cubes to make a model from the building plans below. Record the base plan for your building. What are the maximum and minimum numbers of cubes that could be used to build the structure. 

• Answer: The range of number of cubes is 18-22. 20 cube answer is as under (there are several answers)

• Arrange six people in a pattern so that each of them can see exactly 4 other people. Note that a person in middle can block the view for what is beyond them.
• Answer: