Batch 2 - Class 73 - Coloring Cubes & Spatial Visualization III

Preclass Exercise:

(Geoffrey - 87) A kid got 100% marks in geometry. Her elder sister remarked, "I suppose you think you know geometry from A to Z." "Sure I do." "Prove it, then." The elder sis drew a big A and a big Z, and asked the younger one to divide the A into four parts, which can be put together to make the Z, and without turning over any of the pieces. Can yo do that?

Attendance: Anisha, Smiti, Muskaan, Diya, Adit, Abhiram, Liza, Nandini, Khushi

Class Notes:

In a rectangular room with dimensions 30x12x12, a spider is located in the middle of one 12x12 wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest total distance the spider must crawl along the walls, ceiling, and floor in order to capture the fly?

The answer, 40, can be obtained by "flattening" the walls as illustrated below. Note that his distance is shorter than the 42 the spider would have to travel if first crawling along the wall to the floor, then across the floor, then up one foot to get to the fly.

In this case, flattening in the manner above "loses" a path. Its easier to just go across the ceiling and down. This path is visible in an alternate "flattening" of the room.

Coloring Cubes - Cut, Then Paint

Now we're going to first cut the cube into unit cubes, paint them, and then put them back together. But there's a little catch! We want to paint it with several different colors, in fact as many as possible, so that it can be reassembled to make a cube whose outside is all one color. With a 1-unit cube, this problem is a bit too easy. Paint the cube one color, and it's already back together.
With a 2-unit cube, you can cut it into 8 1-unit cubes, and paint them with two colors in such a way that you could put them back together into a 2-unit cube of either color. How should you paint them?
With a 3-unit cube, cut into 27 1-unit cubes, can you paint them with three colors in such a way that they can be put back together into a 3-unit cube of any of the three colors? If so, how? If not, why not?

All faces and colors have to be used. So any small cube must not have repetition of color unless that combination is going to be used as an edge (2) or corner (3)
Color 18 cubes with 3, 2, and 1 of individual colors
Color 6 cubes with 2, 2, and 2 of individual colors
Color 3 cubes with 3, 3 of individual colors (Note that one can also approach it from here because the center cube in each config must not have that config's color)
Now each cube and its each color can be used independently

Does this generalize? Starting with an n-unit cube, can you cut it into 1-unit cubes and paint them with n colors in such a way that they can be reassembled into an n-unit cube of any one of the colors?

Think about what happens when you get bigger cube compositions? Does the problem become easier or harder?

Using Perspectives

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

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)

Homework

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.

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