Numan Sheikh FC College Lahore

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Liliana West
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3 Five men crash-land their airplane on a deserted island in the South Pacific. On their first day they gather as many coconuts as they can find into one big pile. They decide that, since it is getting dark, they will wait until the next day to divide the coconuts. That night each man took a turn watching for rescue searchers while the others slept. The first watcher got bored so he decided to divide the coconuts into five equal piles. When he did this, he found he had one remaining coconut. He gave this coconut to a monkey, took one of the piles, and hid it for himself. Then he jumbled up the four other piles into one big pile again. To cut a long story short, each of the five men ended up doing exactly the same thing. They each divided the coconuts into five equal piles and had one extra coconut left over, which they gave to the monkey. They each took one of the five piles and hid those coconuts. They each came back and jumbled up the remaining four piles into one big pile. What is the smallest number of coconuts there could have been in the original pile? 3

4 Answer!!!!! Algebra, Discrete Maths, etc!!! 4

to teach a computer or to come up with an algorithm to solve

5 Well of course you can memorize the solution!!! There are plenty available on the internet But in order to teach a computer or to come up with an algorithm to solve it you may need graph theory, group theory, etc. and may be Computer Vision 5

6 The instant insanity game consists of four cubes which are colored differently on each face with the colors red, white (or yellow), blue, and green. The object of the game is given four of these blocks the player is to figure out how to stack the four blocks on top of one another so as to obtain four different colored faces on each side of the stack. There are many different games and many different solutions. 6

in the stack the possibility of stacks is: 24 2 =

7 The problem with this game is in using four blocks which can each be placed in 24 different positions in the stack the possibility of stacks is: 24 2 = 331,776 possible stacks That is a lot of stacks! 7

8 One way is to play around with the cubes for hours until you figure it out. However, there is a math approach using graphs and graph theory 8

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10 I am sure you are all familiar with Tic-tac-toe? No? زریو-اکاٹ In 1976, Frank Harary introduced a new form of tictac-toe. As usual, players alternate X's and O's. 10

11 For a particular shape and board, player X attempts to make the shape, while player O prevents the shape. For example, let us look at the following shape Who wins on a particular board, For example, our familiar 8x8 chess board? No-body knows. What would it take to solve? 11

12 Two Players: I and II A move consists of removing one, two, or three chips from the pile Players alternate moves, with Player I starting Player that removes the last chip wins 12

13 If there are 1, 2, or 3 only, player who moves next wins If there are 4 chips left, player who moves next must leave 1, 2 or 3 chips, and his opponent will win With 5, 6 or 7 chips left, the player who moves next can win by leaving 4 chips 13

14 0, 4, 8, 12, 16, are target positions; if a player moves to that position, they can win the game Therefore, with 21 chips, Player I can win! 14

15 If there is 1 chip, the player who moves next loses If there are 2,3, or 4 chips left, the player who moves next can win by leaving only 1 In this case, 1, 5, 9, 13, are a win for the second player 15

16 There are two players There is a finite set of possible positions The rules of the game specify for both players and each position which moves to other positions are legal moves The players alternate moving The game ends in a finite number of moves (no draws!) 16

17 Normal Play Rule: The last player to move wins Misère Play Rule: The last player to move loses A Terminal Position is one where neither player can move anymore 17

18 No random moves (This rules out games like Poker, Ludo and other dice games) No hidden moves (This rules out games like battleship and a lot of card games) No draws in a finite number of moves (This rules out tic-tac-toe) 18

19 P-Position: Positions that are winning for the Previous player (the player who just moved) N-Position: Positions that are winning for the Next player (the player who is about to move) 19

20 0, 4, 8, 12, 16, are P-positions; if a player moves to that position, they can win the game 21 chips is an N-position 20

21 That means: For any move that N makes There exists a move for P such that For any move that N makes There exists a move for P such that There exists a move for P such that There are no possible moves for N 21

22 P-positions and N-positions can be defined recursively by the following: (1) All terminal positions are P-positions (2) From every N-position, there is at least one move to a P-position (3) From every P-position, every move is to an N-position 22

23 Two-player game, where each move consists of taking a square and removing it and all squares to the right and above. So it is a Misère game!!! Player who takes position (0,0) loses 23

24 N-Positions! 24

25 P-position! 25

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29 Theorem: Proof: The winning strategy for player I is to chomp on (1,1), leaving only an L shaped position Then, for any move that Player II takes, Player I can simply mirror it on the flip side of the L 29

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31 Proof: Theorem: Look at this first move: If this is a P-position, then player 1 wins Otherwise, there exists a P-position that can be obtained from this position And player I could have just taken that move originally 31

32 So we have just learnt that Player I can win in any rectangular starting position of Chomp What about his strategy? What about the strategy of a non-rectangular Chomp This is an Open Question!!! 32

33 Two-player game, where each move consists of taking the token and moving it either downwards or to the left (but not both). Player who makes the last move (to (0,0)) wins Similarities with Chomp? Any? Differences? 33

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35 A Graph or Network is a set of vertices (dots) with edges (lines) connecting them. A B A B A multiple edge D C D C A loop A B D C Two vertices are adjacent if there is a line between them. The vertices A and B above are adjacent because the edge AB is between them. An edge is incident to each of the vertices which are its end points. The degree of a vertex is the number of edges sticking out from it. 35

Lets take a complete graph on 6 vertices. 2 players: green and yellow. All the edges in the graph are initially not colored. Players alternate by coloring an uncolored edge. Green draws green edges.

36 Lets take a complete graph on 6 vertices. 2 players: green and yellow. All the edges in the graph are initially not colored. Players alternate by coloring an uncolored edge. Green draws green edges. Yellow draws yellow edges. If a green triangle is formed, green loses. If a yellow triangle is formed, yellow loses. If all possible edges get drawn without a monochromatic triangle (triangle of a single color) being formed, green and yellow tie. 36

37 Is it possible that the game ends in a tie, even if two players cooperate? The answer is NO, Why? Well we require a proof!!! What about game on complete graph on 5 vertices? What about the game on other graphs? other substructures? more than two players? All are interesting questions, and a lot of them are still open 37

38 B R G W R G G W G B W 1 B 2 G R R W R B R W B G R 3 W 4 Graph Theory Yay!!! 38

39 We will model each cube with a multi-graph. The vertices will correspond to the four colors and we connect the corresponding vertices u and v if there is a pair of opposite faces colored u and v. 39

40 1 Now construct a single multi-graph with 4 vertices and the 12 edges, labeling each edge by the cube associated with it R W B G 2 40

41 Suppose the puzzle has a solution. How would it be represented on the final multi-graph? One sub-graph will represent the front and back of the tower and a second sub-graph will represent the sides of the tower. Using an edge in a sub-graph corresponds to a positioning of the cube (either front/back or sides). 3 4 R W 1 4 B G 2 41

42 What are the restrictions on the sub-graphs and how do they relate to the solution? Uses all four vertices. (all four colors) 42

43 What are the restrictions on the sub-graphs and how do they relate to the solution? Uses all four vertices. Must contain four edges, one from each cube. (all four colors) (orient each cube) 43

44 What are the restrictions on the sub-graphs and how do they relate to the solution? Uses all four vertices. Must contain four edges, one from each cube. No edge can be used more than once. (all four colors) (orient each cube) (can t use same orientation twice) 44

45 What are the restrictions on the sub-graphs and how do they relate to the solution? Uses all four vertices. Must contain four edges, one from each cube. No edge can be used more than once. Each vertex must be of degree 2. (all four colors) (orient each cube) (can t use same orientation twice) (use that color twice, one front & one back, or one left & one right) Remember any similar Graph property??? 45

46 4 f b b f 2 3 b f f 1 b l r r l 1 l r l r Now, stack the cubes using these faces as the front/back and sides. Since each edge represents an orientation, label the edges to determine the orientation. Is there another solution? Is it possible to find a set of sub-graphs that use the loops? 3 4 R B W 1 G

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Plan. Related courses. A Take-Away Game. Mathematical Games , (21-801) - Mathematical Games Look for it in Spring 11

Plan. Related courses. A Take-Away Game. Mathematical Games , (21-801) - Mathematical Games Look for it in Spring 11 V. Adamchik D. Sleator Great Theoretical Ideas In Computer Science Mathematical Games CS 5-25 Spring 2 Lecture Feb., 2 Carnegie Mellon University Plan Introduction to Impartial Combinatorial Games Related