Enjoy Algorithmic Thinking

The sieve of Eratosthenes is an ancient and efficient algorithm for finding all primes numbers from 2 to n. This algorithm finds all the prime numbers in a segment using O(nlog⁡log⁡n) operations.

Given a positive integer n, write a program to check if the number is prime or not. A number n > 1 is said to be a prime number if 1 and n are its only factors. In other words, a prime number is a number that is divisible only by two numbers itself and one.

The birthday paradox is strange and counter-intuitive. It's a "paradox" because our brain find it difficult to handle the compounding power of exponents. Real-world applications for this include a cryptographic attack called the "birthday attack".

There are 25 horses among which we need to find out the fastest 3 horses. In each race, only 5 horses can run simultaneously because there are only 5 tracks. What is the minimum number of races required to find the 3 fastest horses without using a stopwatch?

A group of four people, who have one torch, need to cross a bridge at night. A maximum of two people can cross the bridge at one time, and any party that crosses (either one or two people) must have the torch with them. The torch must be walked back and forth and cannot be thrown. Person A takes 1 minute to cross the bridge, person B takes 2 minutes, person C takes 5 minutes, and person D takes 10 minutes. A pair must walk together at the rate of the slower person’s pace. Find the fastest way they can accomplish this task.

There are n+1 people at a party. They might or might not know each other names. There is one celebrity in the group, and the celebrity does not know anyone by their name. However, all the n people know that celebrity by name. You are given the list of people present at the party. And we can ask only one question from each one of them. “Do you know this name”? How many maximum numbers of questions do you need to ask to identify the actual celebrity?