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question:We need a function for a game scoring system that calculates the possible ways to achieve a certain score using a specified set of scoring plays. Each play can have a different score value (e.g., 1 point, 2 points, etc.), and the function should consider the order of the plays. Create `score_combinations()`, which receives two arguments. ``` score_combinations((1), (2)) ``` ``` (1) - The target score to be achieved (2) - A list of integers representing the possible scores of each play ``` `score_combinations()` should return the total number of different ways the target score can be achieved using any combination of the given plays. Consider the following cases: ```python score_combinations(4, [1, 2, 3]) ------> 7 /// Explanation: We can achieve the score 4 using the following combinations of plays: [1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [2, 1, 1], [2, 2], [1, 3], and [3, 1]. Thus, there are 7 different ways. /// ``` Let's see another example: ```python score_combinations(5, [1, 2]) ------> 8 /// Explanation: We can achieve the score 5 using the following combinations of plays: [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 2, 1], [1, 2, 1, 1], [2, 1, 1, 1], [2, 2, 1], [2, 1, 2], [1, 2, 2]. Thus, there are 8 different ways. /// ``` Challenge yourself, and have fun coding!
answer:def score_combinations(target, plays): Calculates the number of different ways to achieve the target score using the given plays. Parameters: target (int): The target score to be achieved plays (list): A list of integers representing the possible scores of each play Returns: int: The number of different ways to achieve the target score # Initialize a list to store the number of ways to achieve each score from 0 to target dp = [0] * (target + 1) dp[0] = 1 # There's 1 way to achieve the score of 0 (by choosing no play) # Iterate through each possible score from 1 to target for i in range(1, target + 1): # Iterate through each play for play in plays: # If the current score can be reached by adding this play to a previous score if i >= play: dp[i] += dp[i - play] return dp[target]
question:Rafael is organizing a marathon run, and he wants to distribute the water stations in a way that ensures every runner will have access to water within every given interval of kilometers. He wants to know the minimum number of water stations needed so that the distance between two consecutive stations does not exceed d kilometers. Can you help Rafael determine the minimum number of water stations needed? -----Input----- The first line contains two integers n and d (1 ≤ n ≤ 10^5, 1 ≤ d ≤ 10^9), where n is the length of the marathon route (in kilometers) and d is the maximum interval (in kilometers) at which water stations must be placed. -----Output----- Output a single integer, the minimum number of water stations needed. -----Examples----- Input 12 3 Output 4 Input 15 5 Output 3 -----Explanation----- In the first example, the marathon is 12 kilometers long and the maximum interval between water stations is 3 kilometers. The minimum number of water stations needed is 4, and they can be placed at kilometers 0, 3, 6, and 9 (or any other valid configuration meeting the requirement). In the second example, the marathon is 15 kilometers long with a maximum interval of 5 kilometers, hence 3 stations are needed, placed at kilometers 0, 5 and 10 (or any other valid configuration meeting the requirement).
answer:def minimum_water_stations(n, d): Returns the minimum number of water stations needed so that the distance between two consecutive stations does not exceed d kilometers. :param n: total length of the marathon route in kilometers :param d: maximum interval in kilometers between water stations :return: minimum number of water stations # To cover the entire route of n kilometers with intervals not exceeding d kilometers, # we need (n + d - 1) // d stations return (n + d - 1) // d
question:A company organizes a "Secret Santa" event where employees exchange gifts anonymously. Each employee has a preference list of colleagues to whom they would like to give a gift, ranked from 1 to N, where N is the number of employees. The company wants to maximize the total preference ranking value of the pairings. Given the number of employees N and a list of lists of integers representing the preference rankings of each employee, write a program to determine the maximum total preference value achieved by assigning each employee exactly one colleague to give a gift to, such that each employee both gives and receives exactly one gift. Input: - N - Total number of employees (2 ≤ N ≤ 100) - A 2D list of integers where the ith list contains N integers representing the preference rankings of the ith employee, with higher numbers representing stronger preferences. Output: - Output a single integer representing the maximum total preference value that can be achieved with optimal secret Santa pairings. Note that the preference rankings are such that a higher integer value indicates a higher preference (1 being the lowest preference). Example: Input: 3 [[3, 1, 2], [1, 3, 2], [2, 1, 3]] Output: 9 Explanation: - The optimal assignment could be: - Employee 1 gives a gift to Employee 1 (preference value 3) - Employee 2 gives a gift to Employee 2 (preference value 3) - Employee 3 gives a gift to Employee 3 (preference value 3) - The total preference ranking value is 3 + 3 + 3 = 9.
answer:from scipy.optimize import linear_sum_assignment def max_preference_value(N, preferences): Calculate the maximum total preference value for the Secret Santa event. Args: - N (int): Total number of employees. - preferences (list of list of ints): Preference rankings of each employee. Returns: - int: The maximum total preference value. # Convert the preference list to a cost matrix suitable for the linear sum assignment problem cost_matrix = [[-preference for preference in row] for row in preferences] # Use the linear_sum_assignment from scipy.optimize to find the optimal assignment row_ind, col_ind = linear_sum_assignment(cost_matrix) # Calculate the maximum total preference value max_value = -sum(cost_matrix[row][col] for row, col in zip(row_ind, col_ind)) return max_value
question:Given two positive integers n and m, find the greatest common divisor (GCD) of n and m using the Euclidean algorithm. The Euclidean algorithm is based on the principle that the GCD of two numbers does not change if the larger number is replaced by its difference with the smaller number. The Euclidean algorithm can be visualized through the recursive relationship: - gcd(a, b) = gcd(b, a % b), where a % b is the remainder when a is divided by b. - The algorithm terminates when b becomes 0, and the GCD is the value of a at that point. Write a function that implements this algorithm and returns the GCD of the given integers n and m. Example 1: Input: n = 48, m = 18 Output: 6 Explanation: The GCD of 48 and 18 is 6 as 48 = 6 * 8 and 18 = 6 * 3. Example 2: Input: n = 56, m = 98 Output: 14 Explanation: The GCD of 56 and 98 is 14. The series of steps are: 56 % 98 = 56 98 % 56 = 42 56 % 42 = 14 42 % 14 = 0. Example 3: Input: n = 101, m = 10 Output: 1 Explanation: The GCD of 101 and 10 is 1 as they are relatively prime. Example 4: Input: n = 25, m = 5 Output: 5 Example 5: Input: n = 3, m = 7 Output: 1 Constraints: 1 <= n, m <= 10^5
answer:def gcd(n, m): Computes the greatest common divisor (GCD) of two positive integers n and m using the Euclidean algorithm. while m != 0: n, m = m, n % m return n