A Joyful AI Research Journey🌳😊

Definition. We say that two numbers a and b are congruent modulo m if they have the same remainder when divided by m. We denote this by a ≡ b mod m We say integers a and b are "congruent modulo n" if their difference is a multiple of n. For example, 17 and 5 are congruent modulo 3 because 17 - 5 = 12 = 4⋅3. It can be expressed by 17 ≡ 5 mod 3. https://aimath.org/news/congruentnumbers/modulo.html..

a, b = 15, 4 print(a // b, a % b) print(divmod(a, b)) result) 3 3 (3, 3) The divmod(a, b) function is a built-in function in Python that takes two numbers as arguments and returns a tuple containing two values: the quotient (the result of the floor division a // b) and the remainder (a % b). Essentially, it combines the operations of floor division and modulo in a single function.

Question) There is a pile of 11 rocks. Two players take rocks from the pile in turns. In each turn a player can either take one or two rocks from the pile. The player that takes the last rock wins the game. You move first. Answer) You need to play in such a way that you maintain the strategy of leaving a multiple of 3 for the opponent. ex) You Take two rocks Your opponent Take two rocks You Take..

The table below gives a summary of the four selection schemes. Question) We have an unlimited supply of tomatoes, bell peppers and lettuce. We want to make a salad out of 4 units among these three ingredients (we do not have to use all ingredients). The order in which we use the ingredients does not matter. How many different salads we can make? Answer) To solve this problem, we can use the conc..

from itertools import product for t in product('abc', repeat=2): print(*t, sep='', end=' ') Result) aa ab ac ba bb bc ca cb cc This usage of the asterisk (`*`) is known as the unpacking operator in Python. It is used to unpack the elements of the iterable `t`. Here's a breakdown of what's happening: - `product('abc', repeat=2)` generates the Cartesian product of the string `'abc'` with itself, r..

In Python, when working with sets: The union operation is performed using the | operator. The union of two sets is a set containing all the elements that are in either of the two sets. For example, if set1 = {1, 2, 3} and set2 = {3, 4, 5}, then set1 | set2 would result in {1, 2, 3, 4, 5}. The intersection operation is performed using the & operator. The intersection of two sets is a set containi..

Summary. Invariants are important tools for proving impossibility, termination, and various bounds. Invariants may take many forms: numbers, "parity", equations, inequalities. To prove impossibility, one finds a quantity that never changes during a process. To prove that a process terminates in a number of steps, one usually finds a quantity that decreases at every step. Double counting is a met..

ChatGPT, OpenAI Algorithms: An algorithm is a step-by-step procedure or formula for solving a problem. It's about the 'how' – how to perform a task, how to process data, how to solve a particular problem. Algorithms are used for a wide range of purposes in computer science, from data sorting and searching to complex problem-solving in various domains. They are the methods or processes followed t..

Q) There are two equally sized cups: cup 1 contains coffee and cup 2 contains milk. Both cups are half full (we are optimists). Your favorite drink is 1/3 coffee and 2/3 milk. Can you get such a drink in cup 1 by transferring (any amount of) liquid between the two cups? Any amount of your favorite drink would work --- the right proportion is what matters. A) It's impossible to create a drink wit..

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), ..