Batch 3 - Class 136 - Planar Geometry

• Of all the 10000 books at the library, if you select one at random, then there is a 90% chance that it has illustrations. Of all the illustrations in all books, if you select one at random, then there is a 90% chance that it is in color. What is the minimum number of books that must contain colored illustrations?
• Answer: 1. One book can have a lot of colored illustrations, and all other 8999 books can have one uncolored one.

• MartinShCol 5.1 - each of two sides of isosceles triangle is one unit long. Without using calculus, find the length of the third side to maximize the triangle's area
• Answer: Use one equal side as base and now rotate the other equal side to maximize the height - this happens when the other side is perpendicular (and the third side is sqrt(2))
• MartinShCol 5.6 - Regular hexagons are inscribed inside and circumscribed outside a circle. If the smaller hexagon has an area of three square units, what is the area of the larger hexagon?

• Answer: There is a long form solution that can be worked out, where the side of internal equilateral triangle can be computed, and that can be used as radius of the circle to calculate the area of outer hexagon. A simpler solution is to reconstruct the diagram as follows. This establishes a clear 24:18 ratio between area of outer hexagon to internal. The area of outside hexagon is 4.

• Answer: Alternate approach. Consider the inscribed and circumscribed triangles to start with. From (a), it is clear that Outer Triangle is 4 times the area of the Inner triangle. From (b), it is clear that the Inner Hexagon is twice the area of the Inner Triangle. From (c), the Outer Hexagon is 6/9 times the Outer Triangle. Putting these together, the Outer Hexagon must be 4/3 of the Inner Hexagon. Hence 4 units.

• Russian Roulette - You are sitting in a smoke-filled room, your single opponent staring you down. Between you, on a table, there is a six shot revolver, containing exactly one live bullet. Someone has spun the cylinder so neither of you knows whether the next shot will be blank or lethal. Each of you will take one shot at the opponent in turn. The game ends when someone gets shot. Should you go first or second? What is the probability of each person getting killed?
• Answer: Doesn't matter - in the beginning, the bullet has equal probability of being in the odd or even slots, and that determines who dies.

• In the Russian Roulette problem above, what if the cylinder is spun after each shot?
• Answer: Go first. P(1) + P(2) = 1, and P(1) = 5/6*P(2) [or equivalently, P(2) = 1/6+P(1)]. So P(1) = 5/11 and P(2) = 6/11. Here P(1) denotes the probability of first person dying, and P(2) denotes the probability of second person dying.