CS 32 Puzzles, Games & Algorithms Fall 2013

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1 CS 32 Puzzles, Games & Algorithms Fall 2013 Study Guide & Scavenger Hunt #2 November 10, 2014 These problems are chosen to help prepare you for the second midterm exam, scheduled for Friday, November 14, Note: some of these problems are harder than the ones that will appear on the exam. (The problems labeled with a black diamond [ ] are especially challenging.) However, every complete and correct solution that is turned in before you take the midterm will count as one extra credit point added to your midterm exam score, up to a maximum of 15 extra-credit points. You must show all of your work for full credit. And your solutions must be original, that is written using only your own words, with appropriate citations if relevant. EXPLAIN, EXPLAIN, EXPLAIN. GOOD LUCK! 1. A Tower of Hanoi puzzle with 8 disks and three pegs is in the state ( ), i.e., the smallest disk sits on the top of the stack of disks on peg 3, the second smallest disk sit on top of the stack on peg 1,..., the second largest disk sits on the bottom of the stack on peg 3, and the largest disk sits on the bottom of the stack on peg 1. How many moves are required to reach the goal state ( )? Explain On what day of the week will the date 2/22/2222 occur? 3. In a game of bridge 52 cards are dealt to four players: North, South, East, and West. How many different ways can the cards be distributed to each player, so that each hand contains 13 cards? In how many of these deals does North receive all four aces? (Hint: Use the multiplication principle).

2 4. A game of Nim begins in the state ( ). Please list all of the initial moves that would enable the first player to win the game. 5. Nimble is a game that I made up. It is played like Nim, except that the player who plays second has the option of passing on his or her first turn, effectively trading positions with the first player. Consequently, if the game began in the state ( ), and the first player decided to remove say 2 tokens from the last heap, the second player could either remove one or more tokens from any one of the current heaps, or pass forcing the first player to play again. Note that the second player s pass can only be used on his or her first turn. (a) If you were the second player, what would you choose to do in the example provided above, where the current state of the game is ( )? (b) Which player, first, second, or neither, is guaranteed a strategy to win every game of Nimble? Explain.

3 6. One-dimension checkers is played like peg-solitaire in that red tokens can only step or jump to the right, blue tokens to the left, and tokens are only allowed to jump over a token of the opposite color. Moreover, when one player jumps over an opponent s token, that token is captured. Double and multiple jumps are possible, and a player must jump if there is an opportunity to do so. And finally, a player who is not able to move, loses. Rather than develop a strategy for the game, write a function called color-count in racket that returns a list that contains the count of the number of tokens of each color in the given state. Thus (color-count (R E R R E E B B R B)) returns (4 3), indicating 4 red and 3 blue tokens; (color-count (E E E R E E E B B E)) returns (1 2), indicating 1 red and 2 blue tokens.

4 7. Place a unique label at the entrance, goal, and each junction and dead end in the following maze. Then construct an equivalent graph. Make sure that the label of each vertex in your graph matches the label that you assigned to the corresponding feature in the maze. What is the degree of your graph? Is the graph cyclic or acyclic? $

5 8. Apply Trémaux s algorithm for a right bearing mouse in the previous maze: $

6 9. Six coins, labeled 1 through 6, are initially placed in a horizontal grid of seven square cells as shown below Note that the leftmost cell is empty. As in our peg solitaire puzzles, no more than one coin can occupy any given cell. Now, however, each coin can be moved to the right or left, regardless of it shade (or color), either by stepping into the adjacent cell if it is empty, or by jumping over a single coin (of either shade) into the empty cell. Thus, from the initial state only two moves are possible: either the dark coin labeled 1 can be pushed into the empty cell, or the light coin labeled 2 can jump over the 1 into the empty cell. The object of the puzzle is to discover a sequence of legal moves, jumps and steps, that takes the initial state above into the goal state: (a) How many distinct states are reachable from the initial state. (Do not try to realize them: there are too many. Simply try to estimate how many states exist.) (b) Without explicitly solving the puzzle please determine how many total moves (jumps plus steps) are required to reach the goal. (Hint: Although the shades of the tokens have nothing at all to do with the action of the puzzle, the difference between the initial and final shade patterns is an important clue.)

7 10. Consider the two-dimensional version of our peg-solitaire puzzle, where the left configuration illustrates the initial state, and the right, the goal state Here the red tokens can only move either to the right or down, stepping one unit into the empty square, or jumping over a single token of either color. Likewise, the blue tokens can only move either to the left or up. Find the minimum sequence of moves that solves the puzzle. You can express your solution by simply listing a sequence of indices that correspond to the squares from which each token is moved. Thus, the sequence 6, 12, 14 indicates that the red token in square 6 steps into square 9; then the blue token in square 12 jumps over the former red token into square 6, and subsequently, the blue token in square 14 jumps over square 13 into square 12.

8 11. Consider a tower of Hanoi puzzle that contains two disks on four pegs, as shown below. Assume the initial peg (1) is on the left, and the goal peg (4) is on the right. (a) How many states (vertices) belong to this puzzle s state transition graph. (b) Construct the puzzle s state transition graph, and label each vertex with an ordered pair of the form (x, y), where x {1, 2, 3, 4} represents the peg on which the small disk is placed, and y represents the same for the large disk. (c) The degree of a vertex is defined to be the number of edges that are incident to it. The degree of a graph is defined to be the maximum degree of all of its vertices. What is the degree of this graph? (d) What is the length of the shortest solution? How many such solutions exist? 12. A tower of Hanoi puzzle is created having eight disks and four pegs. How many states does this puzzle have? What is the degree of its state-transition graph? (Hint: Don t draw the state-transition graph: it is too large. Instead, just imagine a small neighborhood of the graph.) Explain.

9 13. [ ] A kitchen pantry contains two empty bottles, having capacities of 3 and 5 pints, respectively, and a 1 gallon (8 pint) jug that is full of maple syrup (of course it s Vermont Grade A). A pastry chef would like to fill the 5 pint bottle with exactly 4 pints of syrup (which is needed for a certain recipe), leaving exactly 4 pints in the jug. How can this be done using only the three containers? 1 Note that to ensure exactness, each transfer of fluid from one vessel to another must either completely empty one vessel, or completely fill the other, (or on rare occasions, do both). We can analyze this problem by constructing a state-transition graph. Let the list (x, y, z) denote a state in which x pints are contained in the 3 pint bottle, y pints in the 5 pint bottle, and z pints in the gallon jug. Note that 0 x 3, 0 y 5, and 0 z 8, with x + y +z = 8, and x, y, and z assuming integral values. Since z = 8 x y, each state can be represented as a vertex projected onto the two-dimensional x y-coordinate system: 5 y x (a) Using the notation (x, y) for the projection, the initial state would be (0, 0). What coordinate pair corresponds to the goal state? (b) Now we will complete the state-transition graph. There are two initial moves: (i) pour 3 pints of syrup from the jug into the small container, arriving at the state (3,0), or (ii) pour 5 pints of syrup from the jug into the medium container, arriving at state (0, 5). Represent each transition with an edge on the above graph. Are the transitions directed (one-way) or undirected (reversible)? Now continue, and complete the graph. (c) How many states are accessible? What is the degree of the graph? (d) Can you find a sequence of operations that will satisfy the chef? 1 Someone must have run off with all of the measuring cups!

10 14. Consider a Rubik s cube. Let U denote the move that rotates the upper (yellow) face by 90 in the clockwise direction (as shown). Similarly, let F, R, L, B, and D denote 90 rotations in the clockwise direction of the remaining five faces. We use the shorthand R 2 = RR, and R = R 3. Suppose a cube is scrambled according to the following sequence of operations: R 2 BD F 2 U. Construct a sequence of operations that unscrambles the cube. U B L R F D 15. Consider a Rubik s cube. How many different configurations can be reached by a legal sequence of moves? Explain. Don t forget also to review anagrams, Cretan-style labyrinths, the multiplication principle, Tré maux s and Tarry s algorithms, Alcuin s river-crossing problem, the Missionaries and the Cannibals puzzle, the standard Tower of Hanoi puzzle, one and two-dimensional peg solitaire, etc.

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