# Random thoughts on challenges

The best puzzles fall into two categories. 1. You struggled for quite a while, but you found the solution. You are proud of yourself for persevering. 2. You didn't find the solution, but after seeing the solution you think, "I could have thought of that if I had struggled a little longer" or "that's clever!" This encourages you to struggle more on subsequent problems.

The less useful puzzles are those where you fail to find the solution, and after understanding the solution, you still fail to see why the construction was made. You think, "Good thing I didn't struggle for too long. I would have never thought of this!" Sometimes, this is the result of you not being ready for the problem. Other times, such as with *Langley's problem*, the construction is very hard to see, even for those with all the prerequisite knowledge and experience.

Indexing a challenge is difficult, because it's not enough to know the mathematical tools used to arrive at the solution, you must also know the mathematical maturity of the student seeing the problem. You have to find the appropriate time to introduce the problem to give the student a good amount of struggle without causing them to feel inadequate.