Batch 3 - Class 211 - Frog on Tracks

• What is the probability, if you roll a coin across a chessboard and it comes to rest on the board, that it covers some corner of one of the grid squares? (You can assume that the diameter of the coin is less than the sidelength of a single square on the chessboard.)
• Answer: If the radius of coin is r and side of square on chessboard is s, the coin will cover an intersection only if it is in the quarter circle around that intersection. Any given square has four such quarter circles. So area of coin/ area of square is the probability that the coin covers any intersection.

• Can you "count" an infinity?
• There are countable infinities and there are non-countable infinitiies
• A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time. For example, the set of integers {0,1,−1,2,−2,3,−3,…}{0,1,−1,2,−2,3,−3,…} is clearly infinite. However, as suggested by the above arrangement, we can count off all the integers. Counting off every integer will take forever. But, if you specify any integer, say −10,234,872,306−10,234,872,306, we will get to this integer in the counting process in a finite amount of time. Countably infinite is in contrast to uncountable, which describes a set that is so large, it cannot be counted even if we kept counting forever.
• Infinite rooms hotel - One day the hotel is full with infinite guests (1) One man walks in (2) Infinite number of men walk in
• (1) Move up everyone one room; (2) Move up person in room x to room 2x
• https://www.youtube.com/watch?v=Uj3_KqkI9Zo
• Is the frog problem a countable or uncountable infinite problem? If it is countable, then you can map it to natural numbers, and hence be able to get to any given possibility in a finite amount of time

• Imagine that the frog's starting location was given, and we were given that the frog always jumps right. Can you now capture the frog?
• Can you extend this if the direction could be either left or right?
• Now suppose that the frog's jump size and direction was given, but we didn't know its starting point. Can you now capture the frog?
• Think about how you can combine these solutions into one series of picks, which will cover any given starting point, jump size and direction

• Note that we are not starting from what the starting position and jump size where - since that has infinite possibilities, we can get lost in that thinking. All we are thinking is to cover one possibility in each step. And not miss any possibility along the way.
• Note that there are many patterns of how to map the starting position and jump size to natural numbers. Its not a unique mapping. Any of those mappings would work.

• (Dudeney - 334) On a reduced chessboard of 49 squares, St George wishes to kill the dragon. Can you show how, starting from the central square, he may visit once, and only once, every square of the board in a chain of chess knight moves, and end by capturing the dragon on his last move? There are many ways of doing it so find a pretty design!

• Solution: