Randomize algorithm Tutorial 1 Shenggen Timothy Zheng Phone

Randomize algorithm: Tutorial 1 Shenggen (Timothy ) Zheng Phone NO. : (+420) 773013496 Email: zhengshenggen@gmai. com Office: G 402, Centrum Sumavska 1

How to work on a problem (do research) Ø Once upon a time there was baby king, and the king ruled his Baby Island. Because he was the only baby who knew how to do the following trick: He have two cubes, each cube has six faces, and each face has a digit (0 -9). He knew how to present all days in a month with those cubes. 2014/2/25 2

Ø P 1: Start within your comfort zone. And make it even more comfortable. 2014/2/25 3

Ø P 2: Not too easy, not too hard: pick an interesting challenge within your reach. 2014/2/25 4

Ø P 3: Move away from your desired place, make some mistakes. 2014/2/25 5

Ø P 3: looking from a different angle 2014/2/25 6

Ø P 3: Move away from your desired place, make some mistakes. Look from a different angle, and come back to it. 2014/2/25 7

Ø P 4: Play with it 2014/2/25 8

Ø P 4: connect to other things you know 2014/2/25 9

Ø P 4: Play with it, connect to other things you know, make it your own. 2014/2/25 10

Ø P 1: Start within your comfort zone, make it even more comfortable. Ø P 2: Not too easy, not too hard, pick an interesting challenge within your reach. Ø P 3: Move away from your desired place, make some mistakes. Look from a different angle, and come back to it Ø P 4: Play with it, connect to other things you know, make it your own. 2014/2/25 11

Apple game Ø Consider the following two-player game. The game begins with random n apples on the table. Each round, one player, called Alice, takes no more than k but at least 1 apple out of the table. The second player, called Bob, also takes no more than k but at least 1 apple out of the table. The last one to take apples on the table wins the game. Ø Suppose that Alice and Bob are absolutely smart, what is the probability for Alice to win this game? (Hits: follow the four principles) 2014/2/25 12

Bertrand’s Paradox Ø Determine the probability that a random chord of a circle of unit radius has a length greater than the square root of 3, the side of an inscribed equilateral triangle. 2014/2/25 13

1/2 2014/2/25 14

Ø So, the probability is 1/2. Here is an animation that explains the way we can get to this answer: 2014/2/25 15

1/3 2014/2/25 16

So, the probability that the random chord has a length greater than the square root of 3 is 1/3. Here is an animation that explains the way we can get to this answer: 2014/2/25 17

1/4 2014/2/25 18

What’s the right answer 2014/2/25 19