π² You are stuck in a dungeon with a 20-sided die and need to roll a total of 42 to escape.

π If you roll over 42, the dragon keeps you in the dungeon forever.

β The best number to pick for your total is uncertain, but picking 42 gives you many possibilities to win.

π² The finite size of the die affects the outcome and in the long run, the numbers even out.

π’ The probability of rolling a specific number is one in the average skip length of rolls.

π By using recurrence relations, the probability of rolling a specific number can be calculated based on previous numbers.

π² The probabilities of rolling different numbers on a 20-sided die can be calculated using recursion.

π The probability distribution of rolling the sum of two rolls on a 20-sided die forms an interesting graph.

π The best total to give to the dragon in this game is 34, based on the calculated probabilities.

π Numbers around 33 to 35 are likely to come up when picking numbers in a small interval.

π² Between 1 and 20, the number 20 is the most likely to come up in sums, followed by 34.

π’ For a larger die, the sweet spot for the best probability is somewhere between N and 2N, with e being the leading order.

π The rule of thumb for finding the maximum of a function involving natural logarithms and large values of N is to multiply N by a certain approximation.

π² There are continuous versions of the problem involving a die that can roll any number, which result in a smooth distribution of outcomes.

π An evil number is one whose decimal expansion, when the digits are added up, equals 666. Pi is an example of an evil number.

π’ Roughly every fifth number is considered evil.

π² The concept of evil numbers can be related to rolling a 9-sided die.

π There are different definitions of evil numbers based on prime numbers and binary expansion.

π’ There are numbers called 'beastly primes' that contain the number 666 in a special place.

π‘ Brilliant is a platform that offers courses and quizzes on various subjects, including math, logic, data science, and computer programming.

π± Brilliant's courses and quizzes are accessible on mobile devices and provide a seamless, interactive learning experience.

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