The following puzzles were inspired by John's **Puzzle 104** and **Puzzle 105** in [Lecture 36](https://forum.azimuthproject.org/discussion/2204/lecture-36-categories-from-graphs).

**Puzzle MD 1**:

From the New York Times (2013)

> *There are 10 steps in front of you that you are about to climb. At any point, you can either take one step up, or you can jump two steps up. In how many unique ways can you climb the 10 steps?*

**Puzzle MD 2**: Suppose there were \\(10^7\\) steps in front of you. Let \\(N\\) be the number of unique ways you can go up these steps. What are the last 10 digits of \\(N\\)?

(\\(\star\\)) What about the last 10 digits after \\(10^{100}\\) steps?

**Puzzle MD 3**: Suppose you could either go up the stairs 3 different ways: using your left foot to go up one, your right foot to go up one, or hop up two at a time. There's still \\(10^7\\) steps in front of you, what are the last 10 digits of the count now?

(\\(\star\\)) What about the last 10 digits after \\(10^{100}\\) steps?

**Puzzle MD 4**: Now suppose you can go either 1 step, 2 steps, or 3 steps at a time. Once again, what are the last 10 digits of the count for \\(10^7\\) steps?

(\\(\star\\)) What about the last 10 digits after \\(10^{100}\\) steps?

**Puzzle MD 5**: Finally, assume you can go up either 1 step, 2 steps, or two different ways of going 5 steps, or three different ways of going 9 steps. What are the last 10 digits for the count for \\(10^7\\) steps?

(\\(\star\\)) What about the last 10 digits after \\(10^{100}\\) steps?