Exploring Dice Probabilities and Evil Numbers in Relation to Pi

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

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

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

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

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

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

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

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

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

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

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

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

🔑 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.

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

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

🔢 Roughly every fifth number is considered evil.

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

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

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

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