Batch 2 - Class 269 - Spatial Visualization - Coloring Cubes

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Preclass Exercise:

MartinShCol 5.5 - From one corner of a square, extend two lines that exactly trisect the square's area - Into what ratios do these lines cut the two sides of the square?

Answer: 1/3 of each side. If you look at the diagonal originating from trisecting vertex, and look at one half of the square resulting from such a diagonal, the areas within that must be 1:2 ratio. However, the base of both those triangles is the same, so the height must be in 1:2 ratio.

Attendance: Raghav, Aneesh, Aarkin, SiddhantJ, Rehaan, Kabir, Advay, Adyant, Shikhar, Ayush, Kushagra, Vivaan

Class Notes: (Repeat from class 71-72)

Coloring Cubes - Paint, then Cut

We'll take a cube whose edges are n units long; paint its surface completely; cut it into unit cubes, whose edges are 1 unit long; and then ask how many unit cubes are painted in which way. Clearly, if you start with a 1-unit cube, you end up with 1 unit cube painted on all 6 sides, and that's it.
If you start with a 2-unit cube, how many of the resulting unit cubes are completely unpainted? Painted on just 1 side? Two sides? Three sides?
Starting with a 3-unit cube, answer the same questions.
Repeat for a 4-unit cube.
Repeat for an n unit cube. Can you find the pattern?

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?

Homework:

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?

References:

The Colossal Book of Short Puzzles and Problems, by Martin Gardner