Batch 3 - Class 200 - Coloring Cubes

• 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)

• 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? 

• 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?

• 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 2 opposite 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

• A 10x10 square is decomposed into exactly 75 squares of various integer sizes. How many 3x3 squares are there in the decomposition?
• Answer: 2. One way to think about it is that any 2x2 square eats up 3 unit squares (it has 4 squares, and 1 would anyway be there in a 1x1 square) over a unit square, 3x3 eats up 8, 4x4 eats up 15, 5x5 eats up 24 and so on. The total "eat count" must be 25. This means that there are no 4x4 or bigger squares, because then no combination of larger squares eats up exactly 25 squares. It also means there must be 2 3x3 squares, because with other values, we can't add up enough 2x2 squares to total exactly 25 "eaten" squares.

• Consider a n x n x n cube, which is painted on the exterior and then cut into n^3 unit cubes. 
• Find the number of cubes with at least one face painted, as function of n. Note that for n-3, 26 cubes have at least one face painted
• Answer: n^3 - (n-2)^3
• Suppose that each pair of interior faces are glued together so that each pair of faces requires one unit of glue. How many units are required in all? Examine for n=2..6
• Answer: For n=2, 12. For n=3, 54. Whats the pattern? Note that total faces are 6n^3. Of which 6n^2 will finally be exposed. So glued faces are 6n^2(n-1). Since two faces are glued together, the units of glue required are 3n^2(n-1)
• In how many ways can you paint a cube, each side with a different color, excluding rotations
• Answer: 4!+3! (Take color one C1 on top. Another chosen color C2 can be on a side, in which case the other colors have 4! positions. If C2 is at bottom, then fix C3 on right, and that gives another 3! positions)