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Easy, Hard, and Impossible

Easy, Hard, and Impossible. Elaine Rich. Easy. Tic Tac Toe. Hard. Chess. The Turk. Unveiled in 1770. Searching for the Best Move. A B C D E F G H I J K L M

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Easy, Hard, and Impossible

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  1. Easy, Hard, and Impossible Elaine Rich

  2. Easy

  3. Tic Tac Toe

  4. Hard

  5. Chess

  6. The Turk Unveiled in 1770.

  7. Searching for the Best Move A B C D E F G H I J K L M (8) (-6) (0) (0) (2) (5) (-4) (10) (5)

  8. How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358

  9. How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358 2,000,000,000,000 • Seconds in a year 

  10. How Much Computation Does it Take? • Middle game branching factor  35 • Lookahead required to play master level chess  8 • 358 2,000,000,000,000 • Seconds in a year  31,536,000 • Seconds since Big Bang  300,000,000,000,000,000

  11. The Turk Still fascinates people.

  12. How Did It Work?

  13. A Modern Reconstruction Built by John Gaughan. First displayed in 1989. Controlled by a computer. Uses the Turk’s original chess board.

  14. Chess Today In 1997, Deep Blue beat Garry Kasparov.

  15. Seems Hard But Really Easy

  16. Nim At your turn, you must choose a pile, then remove as many sticks from the pile as you like. The player who takes the last stick(s) wins.

  17. Nim Now let’s try:

  18. Nim Now let’s try: Oops, now there are a lot of possibilities to try.

  19. Nim http://www.gamedesign.jp/flash/nim/nim.html

  20. Binary Numbers

  21. Nim My turn: 10 (2) 10 (2) 11(3) 11 To form the last row: XOR each column. XOR: if number of 1’s is even: 0 if number of 1’s is odd: 1

  22. Nim My turn: 10 (2) 10 (2) 11(3) 11 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

  23. Nim My turn: 100 (4) 010 (2) 101 (5) 011 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

  24. Nim Your turn: 100 (4) 001 (1) 101 (5) 000 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise.

  25. Following Paths

  26. Seven Bridges of Königsberg

  27. Seven Bridges of Königsberg Seven Bridges of Königsberg: 1 3 4 2

  28. Seven Bridges of Königsberg Seven Bridges of Königsberg: 1 3 4 2 As a graph:

  29. Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783

  30. Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783 There is a circuit if every node touches an even number of edges.

  31. So, Can We Do It? 1 3 4 2 As a graph:

  32. Unfortuntately, There Isn’t Always a Trick Suppose we need to visit every node exactly once.

  33. The Traveling Salesman Problem 15 25 10 28 20 4 8 40 9 7 3 23 Find the shortest circuit that visits every city exactly once.

  34. Visting Nodes Rather Than Edges ● A Hamiltonian path: visit every node exactly once. ● A Hamiltonian circuit: visit every node exactly once and end up where you started. All these people care: • Salesmen, • Farm inspectors, • Network analysts

  35. The Traveling Salesman Problem 15 25 10 28 20 4 8 40 9 7 3 23 Given n cities: Choose a first city n Choose a second n-1 Choose a third n-2 … n!

  36. The Traveling Salesman Problem Can we do better than n! ● First city doesn’t matter. ● Order doesn’t matter. So we get (n-1!)/2.

  37. The Growth Rate of n!

  38. Putting it into Perspective

  39. Is This The Best We Can Do? Probably. Would you like to win $1M? The Millenium Prize

  40. Impossible

  41. An Interesting Puzzle 2 List 1 b a b b b List 2 b a

  42. An Interesting Puzzle 2 1 List 1 b a b b b b List 2 b a b b b

  43. An Interesting Puzzle 2 1 1 List 1 b a b b b b b List 2 b a b b b b b b

  44. An Interesting Puzzle 2 1 1 3 List 1 b a b b b b b b a List 2 b a b b b b b b a

  45. The Post Correspondence Problem

  46. The Post Correspondence Problem

  47. The Post Correspondence Problem

  48. Can A Program Do This? Can we write a program to answer the following question: Given a PCP problem, decide whether or not it has a solution. Return: True if it does. False if it does not.

  49. The Post Correspondence Problem A program to solve this problem: Until a solution or a dead end is found do: If dead end, halt and report no. Generate the next candidate solution. Test it. If it is a solution, halt and report yes. So, if there are say 4 rows in the table, we’ll try: 1 2 3 4 1,1 1,2 1,3 1,4 1,5 2,1 …… 1,1,1 ….

  50. Will This Work? • If there is a solution: • If there is no solution:

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