Project Euler problem 227 - "The Chase" - is a nice example of a problem where the brute force approach is completely reasonable to reach for first, completely infeasible to actually finish, and directly points you at the right mathematical reformulation.

This is what we ended up with. Our full working notes are on the wiki at Project Euler/227.

The Problem

An even number of players sit around a table. Two players sitting directly opposite each other each start with a die. Each round, both players roll:

• Roll a 1 → pass the die to your left neighbor
• Roll a 6 → pass …

Here is a nice little problem from Paul Nahin's Will You Be Alive 10 Years From Now?. It is a problem from the 1654 correspondence between Antoine Gombaud (the Chevalier de Méré) and Blaise Pascal - one of the founding episodes of probability theory, dressed up as a gambling puzzle.

(This one lives on our wiki as FMM21, part of our Friday Morning Math Problem series.)

The Two Games

Game 1. You toss a single die \(N_1\) times. What should \(N_1\) be to make the probability of seeing at least one 6 greater than \(\tfrac{1}{2}\)?

Game 2. You toss …