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 …

Project Euler problem 60 is a nice mid-difficulty combinatorial search problem. It has enough structure that pure brute force falls over, and enough structure that a handful of small observations gives you a very tractable search.

Our working notes: Project Euler/60.

The Problem

The primes 3, 7, 109, and 673 are "quite remarkable" - concatenating any two of them in either order gives another prime. So 7109, 1097, 3673, 6733, 109673, 673109, and so on, are all prime. That is 12 concatenations, all prime.

The sum of these four primes is 792, and the problem states this is the smallest …