# Function Generating Problems

Imagine you’re at a party and you’re asked to put together a playlist for the evening. You know that there are 10 songs you absolutely want to play, but the order matters. Play the wrong song at the wrong time, and you might kill the party mood.

Function generating problems are like figuring out the number of unique playlists you can create with these 10 songs. You might say, “Well, that’s simple, it’s just 10 factorial (10!).”

But then, let’s make it more complicated. Some songs pair really well together and should be played back-to-back, while others should never be played one after the other. So, not every sequence is acceptable.

This problem of generating functions that count the possible ways to arrange things (like our songs), according to certain rules, is common in computer science, particularly in the field of combinatorics.

You might construct a function, for example, that takes the number of songs (n) and the restrictions on the playlist, and it outputs the number of possible playlists. This function is a “function generator,” a tool that helps you to explore the many possible solutions to a problem according to the rules or constraints you’ve defined.

So, just like making a party playlist that keeps everyone dancing, function generating problems help us understand the myriad ways we can arrange or combine components within defined parameters to reach a desired outcome.

A type of problem that lends itself to systematic externalization of information involves determining a general function for a large number of events. The plan is to look at the results for small numbers of corresponding events and to see if a pattern emerges.

### Climbing Staircase Problem

Function generating problems, as the name implies, require us to find a general function that describes the outcome of a large set of events. This typically involves observing results for smaller subsets of events and detecting any emerging patterns. A prime example of a function generating problem is the “Climbing Staircase” problem.

The Climbing Staircase problem poses the following scenario: You have a staircase with ’n’ steps, and you can ascend by either 1 or 2 steps at a time. The question is, how many different ways can you climb the staircase?

To derive a general function from this problem, let’s examine the scenarios for smaller numbers of steps:

If there’s only one step, there’s just one way to climb itâ€”by taking that single step. So, for n=1, the number of ways is 1.

If there are two steps, there are two possibilities: you can climb each step individually (2 steps of 1), or you can take a single leap (1 step of 2). So, for n=2, the number of ways is 2.

If there are three steps, you can climb them in the following ways: 1-1-1, 1-2, or 2-1. So, for n=3, the number of ways is 3.

On first glance, it might seem like the number of ways to climb the stairs is simply equal to the number of steps. But does this pattern hold for larger staircases? Let’s consider a staircase with four steps:

- 1-1-1-1
- 1-1-2
- 1-2-1
- 2-1-1
- 2-2

Now, we can see that for n=4, the number of ways is 5, not 4. This indicates that our initial observation doesn’t hold for all cases, and a different pattern is at play.

It turns out that the number of ways to climb ’n’ steps is equal to the sum of the number of ways to climb ’n-1’ steps and ’n-2’ steps. This pattern is similar to the Fibonacci sequence, where each term is the sum of the previous two. Here, for each additional step, you can reach it either from the previous step or the step before that.

Function generating problems like this one necessitate careful observation and pattern detection. By analyzing simpler cases, we can often uncover the rules governing more complex scenarios.