Consider a tournament between N teams, each team playing each of the other teams. Show (by example) there is a tournament that might occur, where every team is beaten by some team.
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Consider a tournament between N teams, each team playing each of the other teams. Show (by example) there is a tournament that might occur, where every team is beaten by some team.
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- This problem is taken from the delightful book "Problems for Mathematicians, Young and Old" by Paul R. Halmos. Suppose that 931 tennis players want to play an elimination tournament. That means: they pair up, at random, for each round; if the number of players before the round begins is odd, one of them, chosen at random, sits out that round. The winners of each round, and the odd one who sat it out (if there was an odd one), play in the next round, till, finally, there is only one winner, the champion. What is the total number of matches to be played altogether, in all the rounds of the tournament? Your answer: Hint: This is much simpler than you think. When you see the answer you will say "of course".Imagine there are N teams competing in a tournament, and that each team plays each of the other teams once. If a tournament were to take place, it should be demonstrated (using an example) that every team would lose to at least one other team in the tournament.There is an upcoming football tournament, and the n participating teams are labelled from 1 to n. Each pair of teams will play against each other exactly once. Thus, a total of matches will be held, and each team will compete in n − 1 of these matches. There are only two possible outcomes of a match: 1. The match ends in a draw, in which case both teams will get 1 point. 2. One team wins the match, in which case the winning team gets 3 points and the losing team gets 0 points. Design an algorithm which runs in O(n2 ) time and provides a list of results in all matches which: (a) ensures that all n teams finish with the same points total, and (b) includes the fewest drawn matches among all lists satisfying (a). Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English.Please give time complexity. list of results mean Any combination of wins, losses and draws. You may wish to view this as a mapping from the set of…
- In a tournament, there are n participating teams are labelled from 1 to n. Each pair of teams will play against each other exactly once. Thus, a total of [n(n-1)/2] matches will be held, and each team will compete in n − 1 of these matches. There are only two possible outcomes of a match: 1. The match ends in a draw, in which case both teams will get 1 point. 2. One team wins the match, in which case the winning team gets 3 points and the losing team gets 0 points. Design an algorithm which runs in O(n2 ) time and provides a list of results in all [n(n-1)/2] matches which: (a) ensures that all n teams finish with the same points total, and (b) includes the fewest drawn matches among all lists satisfying (a). Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain EnglishIn a tournament, there are n participating teams are labelled from 1 to n. Each pair of teams will play against each other exactly once. Thus, a total of [n(n-1)/2] matches will be held, and each team will compete in n − 1 of these matches. There are only two possible outcomes of a match: 1. The match ends in a draw, in which case both teams will get 1 point. 2. One team wins the match, in which case the winning team gets 3 points and the losing team gets 0 points. Design an algorithm which runs in O(n2 ) time and provides a list of results in all [n(n-1)/2] matches which: (a) ensures that all n teams finish with the same points total, and (b) includes the fewest drawn matches among all lists satisfying (a). Do not write the code, give steps and methods. Explain the steps of algorithm, and the logic behind these steps in plain English. PLease give the total time complexity.Dingyu is playing a game defined on an n X n board. Each cell (i, j) of the board (1 2, he may only go to (2, n).) The reward he earns for a move from cell C to cell D is |value of cell C – value of cell D|. The game ends when he reaches (n, n). The total reward - is the sum of the rewards for each move he makes. For example, if n = 1 2 and A = 3 the answer is 4 since he can visit (1, 1) → (1, 2) → (2, 2), and no other solution will get a higher reward. A. Write a recurrence relation to express the maximum possible reward Dingyu can achieve in traveling from cell (1, 1) to cell (n, n). Be sure to include any necessary base cases. B. State the asymptotic (big-O) running time, as a function of n, of a bottom-up dynamic programming algorithm based on your answer from the previous part. Briefly justify your answer. (You do not need to write down the algorithm itself.)
- Let's revisit our first problem, where we want to set up a series of chess matches so we can rank six players in our class. As we did before, we will assume that everyone keeps their chess rating a private secret; however, when two players have a chess match, the person with the higher rating wins 100% of the time. But this time, we are only interested in identifying the BEST of these six players and the WORST of these six players. (We don't care about the relative ordering or ranking of the middle four players.) Your goal is to devise a comparison-based algorithm that is guaranteed to identify the player with the highest rating and the player with the lowest rating. Because you are very strong at Algorithm Design, you know how to do this in the most efficient way. Here are five statements. A. There exists an algorithm to solve this problem using 6 matches, but there does not exist an algorithm using only 5 matches. B. There exists an algorithm to solve this problem using 7 matches,…A group of people, numbered 1 to N, are sitting in a circle. Starting at person 1, a hot potato is passed. After x number of passes, the person holding the hot potato is eliminated, the circle closes ranks, and the game continues with the person who was sitting after the eliminated person picking up the hot potato. The last remaining person wins. For example: number of passes = 1 and number of players = 5, the order of elimination is 2, 4, 1, 5. Write a program for general values of X and N. Ask a user for the number of people and number of passes To speed up the input and debugging, you may wish to store the names of the people in a file. Make sure no two names start the same letter ( Alex and Ana are not OK). Alternatively, you can number the players. Output number and/or the name of a person being eliminated Output number and the name of the winner Do not expect a user to do the right thing, error check the user input; among other things, what do you think a reasonable…A group of people, numbered 1 to N, are sitting in a circle. Starting at person 1, a hot potato is passed. After x number of passes, the person holding the hot potato is eliminated, the circle closes ranks, and the game continues with the person who was sitting after the eliminated person picking up the hot potato. The last remaining person wins. For example: number of passes = 1 and number of players = 5, the order of elimination is 2, 4, 1, 5. Write a program for general values of X and N. Ask a user for the number of people and number of passes To speed up the input and debugging, you may wish to store the names of the people in a file. Make sure no two names start the same letter ( Alex and Ana are not OK). Alternatively, you can number the players. Output number and/or the name of a person being eliminated Output number and the name of the winner Do not expect a user to do the right thing, error check the user input; among other things, what do you think a reasonable…
- Given a deck of 52 playing cards, we place all cards in random order face up next to each other. Then we put a chip on each card that has at least one neighbour with the same face value (e.g., on that has another queen next to it), we put a chip. Finally, we collect all chips that were each queen placed on the cards. For example, for the sequence of cards A♡, 54, A4, 10O, 10♡, 104, 94, 30, 3♡, Q4, 34 we receive 5 chips: One chip gets placed on each of the cards 100, 10♡, 104, 30, 3♡. (a) Let p5 be the probability that we receive a chip for the 5th card (i.e., the face value of the 5th card matches the face value of one of its two neighbours). Determine p5 (rounded to 2 decimal places). (b) Determine the expected number of chips we receive in total (rounded to 2 decimal places). (c) For the purpose of this question, you can assume that the expectation of part (b) is 6 or smaller. Assume that each chip is worth v dollars. Further, assume that as a result of this game we receive at least…In the game of 10-pin bowling the bowler has two attempts to knock down pins for every frame of 10-pins, and scores a point for each pin knocked down. If all the pins are knocked down with two attempts, the bowler gets a bonus - whatever score they obtain with their next bowl is doubled. If all the pins are knocked down on the 1st attempt, no 2nd attempt is allowed, and the bowler gets a bonus – whatever score they obtain on their next two bowls are doubled. A student attempts to capture this scoring system in VHDL code, a fragment of which is shown in Figure Q4. Q4 (a) Draw the state transition diagram described by the VHDL of Figure Q4. Discuss whether the VHDL of Figure Q4 correctly scores a game of 10-pin bowling. (b) elsif CLK='1' and CLK'event and UPD='1' then case present state is when throwl => frame := frame + 1; + resize (unsigned (N), 9) score := score if N = w1010" then present state score := score + resize (unsigned (N), 9) present state + resize (unsigned (N),9) if N =…On a chess board of r rows and c columns there is a lone white rook surrounded by a group of opponent's black knights. Each knight attacks 8 squares as in a typical chess game, which are shown in the figure - the knight on the red square attacks the 8 squares with a red dot. The rook can move horizontally and vertically by any number of squares. The rook can safely pass through an empty square that is attacked by a knight, but it must move to a square that is not attacked by any knight. The rook cannot jump over a knight while moving. If the rook moves to a square that contains a knight, it may capture it and remove it from the board. The black knights. never move. Can the rook eventually safely move to the designated target square? The figure illustrates how the white rook can move to the blue target square at the top-right corner in the first sample case. The rook captures one black knight at the bottom-right of the board on its way. Rok nd kight lcoes by Chunen Input The first line…