Batch 2 - Class 71 - Finding Toilets, Coloring Cubes & Spatial Visualization

• Ten slips of paper, numbered 1 through 10, are placed in a hat. Three numbers are drawn out, one after another. What is the probability that the three numbers are drawn in increasing order?
• Answer: 1/6

• Normally when you go to a music festival or a cricket match, you may see a lot of toilets. However, most of them are not clean. How will you choose a toilet without having to look at all of them? 

  Rules: If you have checked a toilet and discarded it, you can't go back to it because there is a queue building up. You can, however, go to the next (whichever you decide is "next") toilet and check that, and so on.Explain the notion that lets assume all toilets have different levels of cleanliness, so its all relative - there is no absolute benchmark of cleanliness. For example, we can number the toilets 1, 2, 3... by cleanliness once we have seen all of them
• Let kids come up with different ways and calculate the chances of them picking up the best toilet as per those algorithms

  Notice that looking at all toilets is not a good option, because then you have to pick the last one
• What if we saw the first, rejected it, and then looked on for a better toilet than that

  List all cases, the chances of picking the best toilet are 1/2 - better than if we just picked the first toilet!
• Lets try this with 4 toilets (Answer 11/24 probability of finding the first toilet)

  What if we checked first two and rejected them, and then picked the better one after that?Probability is dropping with "1 reject"Any ideas of how we maximize the best toilet, even as number of toilets get large?Answer: Reject about 37% and then pick the better one after that. Now you will have about a 37% chance of picking the right toilet (Actually this number is 1/e, for the more mathematically inclined)So there is still a way of picking the best toilet more than a third of the time, even if number of toilets is very large, but you may miss the festival checking out 37% of the toilets first!

• 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? 

• 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)