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Q) In a group of 27 students every girl knows four boys and every boy knows five girls. Find the number of boys in the group. A) To solve this problem, let's denote the number of boys in the group as B and the number of girls as G. We know that the total number of students in the group is 27, so we can express the relationship between the number of boys and girls as: G+B=27 We also know that eac..

"Recursion" is a way of defining some mathematical object (including a function or computation whose definition involves a recursive algorithm); "Induction" is a way of proving some mathematical statement. Extremely often, if a mathematical statement is made about a recursively-defined object, then the proof of that statement will involve induction. https://math.stackexchange.com/questions/37792..

Problem) There are n individuals at a party, and some of them have shaken hands. Prove that there are two individuals who have participated in the same number of handshakes. Solution) To prove that there are two individuals who have participated in the same number of handshakes at a party with n individuals, we can use the pigeonhole principle. First, let's consider the possible number of handsh..

Problem) Does there exist a power of 2 that starts with 65? Solution) for n in range (100): if int(str(2 ** n) [:2]) == 65: print（f'2**｛n｝=｛2**n｝'） In the statement for n in range(100):, the colon : is used to indicate the start of the block of code that will be executed in each iteration of the for loop. It is a fundamental part of Python's syntax for defining loops, conditionals, and other con..

OpenAI, ChatGPT Researchers are driven by a variety of motivations, which can vary depending on their personal interests, career goals, and the nature of their work. Common motivations for researchers include: Intellectual Curiosity: Many researchers are inherently curious and have a strong desire to explore unknowns, solve puzzles, and understand complex phenomena. Advancement of Knowledge: A f..

Key points about mathematical induction: Mathematical induction is used to prove that some statements A(i) hold for all values of i. An induction proof consists of two parts: the base case and the induction step. The base case assures that A(i) holds for some (not necessarily small) values of i. The base case and the induction step must be consistent: if the induction step uses A(n) and A(n−1), ..

Develop a Python method change(amount) that for any integer amount in the range from 24 to 1000 returns a list consisting of numbers 5 and 7 only, such that their sum is equal to amount. For example, change(28) may return [7, 7, 7, 7], while change(49) may return [7, 7, 7, 7, 7, 7, 7] or [5, 5, 5, 5, 5, 5, 5, 7, 7] or [7, 5, 5, 5, 5, 5, 5, 5, 7]. To solve this quiz, implement the method change(a..

Modify the code considered in the lectures to find out the number of solutions to the 8 queens puzzle. For example, as was explained in the videos, for the 4 queens puzzle there are 2 solutions. import itertools as it def is_solution(perm): for (i1, i2) in it.combinations(range(len(perm)),2): if abs(i1-i2)==abs(perm[i1]-perm[i2]): return False return True assert(is_solution([1,3,0,2]) == True) a..

Question There are some books on the table. If you group them by 3, you get some number of full groups and 2 books remain; if you group them by 4, you get some number of full groups and 3 books remain; if you group them by 5, you get some number of full groups and 4 books remain. What is the number of books on the table, if it is less than 100? Solution Using the Chinese Remainder Theorem: N (Pr..

Question Person A has an unlimited number of 7-florin coins, person B has an unlimited number of 13-florin coins. How A can pay 5 florins to B? 7x - 13y = 5 2*A - B = D 2*7 - 13 = 1 So let's find a multiple to get our goal and it's 5 because 5 x D or (5x1) is 5. Now let's multiply each value. 14 - 13 = 1 14*5 - 13*5 = 5 70 - 65 = 5 x = 70 / 7 = 10 y = 65 / 13 = 5 ∴ x = 10, y = 5