Hector's Puzzled

It’s not quite “Waterworld” yet, but the only land humanity has left is a bunch of circular islands where each continent used to be. Fortunately, we have some good engineers. We are planning to connect the islands to each other through a network of straight-line bridges. But, of course, there are rules: The bridges cannot cross each other or pass over islands, and can only be placed horizontally (east-west) or vertically (north-south). (Diagonal bridges are too difficult for your engineers to build.) A pair of islands can’t be connected by more than two bridges. All the islands must be connected together in a single, contiguous group. In other words, the people on any given island need to be able to reach every other island, even if they have to take a circuitous route. Each island (represented by a circle on the diagrams below) has a value equal to the total number of bridges that connect it to its neighbors. To connect the islands appropriately, you’ll need to follow the diamond-shaped signs sprinkled throughout this dystopia. Each corner of the sign corresponds to one of the four directions (north, south, east, west), and the numbers in those corners show the sum of the values of the islands that lie on a straight line extending in that direction. No value can be repeated within a single sum — for example, if the sum is 10, the values that make it up cannot be 5 and 5. The bridges cannot pass through the signs. One sign on each continent is no longer displaying numbers, and nobody remembers the whole layout anymore. How are we going to build the bridges? [Solution]

Each of seven dudes sleeps in his own bed in a shared dormitory. Every night, they retire to bed one at a time, always in the same sequential order, with the youngest dude retiring first and the oldest retiring last. On a particular evening, the youngest dude is in a jolly mood. He decides not to go to his own bed but rather to choose one at random from among the other six beds. As each of the other dudes retires, he chooses his own bed if it is not occupied, and otherwise chooses another unoccupied bed at random. What is the probability that the oldest dude sleeps in his own bed? What is the expected number of dudes who do not sleep in their own beds? [Solution]

A frog needs to jump across 20 lily pads. He starts on the shore (Number 0) and ends precisely on the last lily pad (Number 20). He can jump one or two lily pads at a time. How many different ways can he reach his destination? What if he can jump one, two or three at a time? Or four? Five? Six? Etc. [Solution]

On the brilliant and ageless game show “The Price Is Right,” there is an important segment called the Showcase Showdown. Three players step up, one at a time, to spin an enormous, sparkling wheel. The wheel has 20 segments at which it can stop, labeled from five cents up to one dollar, in increments of five cents. Each player can spin the wheel either one or two times. The goal is for the sum of one’s spins to get closer to one dollar than the other players, without going over. (Any sum over a dollar loses. Ties are broken by a single spin of the wheel, where the highest number triumphs.) For what amounts should the first spinner stop after just one spin, assuming the other two players will play optimally? [Solution]

You must build a very specific tower out of four differently colored pieces that can be stacked in any order. But when you start building, you don’t know what the correct order is. Upon assembling the pieces in some order, you can consult an architectural oracle (he goes by Frank) who will inform you if zero, one, two or all four pieces of the tower are in the correct position. Your tower doesn’t count as finished until the oracle confirms your solution is correct. How many times should you have to consult the oracle, in the worst case, to assemble the tower correctly? [Solution]