# Art of Problem-Solving: Leveraging Abstract Models for Effective Solutions

A model is an abstract representation of a problem. It is simpler than the real system. A model only includes important aspects of the real problem. The limitations of the model must be understood. Including too little detail in the model might miss relevant interactions and the model will not promote understanding. Including too much detail in the model might make the model complicated and make understanding difficult.

A good example of how the model can help to solve a problem is creating the diagram for the problem: Buy and Sell Stock with Cooldown. In this case, the visual model of the problem is a state diagram that consists of all the states and the transitions. A good model will result in concise and elegant code.

## Translating Complex Systems into Simplified Visual Representations

### For Clearer Understanding and Effective Solution Design

Let’s delve deeper into the topic:

**Abstract Representation**: A model, by definition, is an abstract and simplified representation of a more complex real-world problem or system. This abstraction process helps us to focus on the critical components of the problem and disregards the non-essential details. For example, when modeling an airplane for aerodynamic analysis, we might not need to consider the color of the airplane.**Selective Inclusion**: The key in creating a good model is to know what to include and what to exclude. You only want to include the important aspects of the real problem. Deciding what is ‘important’ depends on the problem at hand and requires a deep understanding of the system. Ignoring vital aspects can lead to a loss in understanding, while including too much can over-complicate the model, making it unwieldy to work with.**Understanding Limitations**: A model is inherently a simplification of reality, so it won’t capture everything perfectly. It’s crucial to understand these limitations and bear them in mind when using the model to draw conclusions or make decisions.**Visual Models and Problem-Solving**: A visual model, such as a flowchart or a state diagram, can be incredibly useful when trying to solve a problem. This visual representation can make it easier to understand the problem and can often directly inform the solution.

## Buy and Sell Stock with Cooldown

In this problem, a trader can make as many transactions as they like but after selling a stock, they must wait for one day before making the next transaction. To model this, we can create a state diagram where each state represents a possible action (buy, sell, or cooldown) and the transitions between states represent possible state changes.

The visual model helps to identify the state transitions and conditions under which these transitions occur. This understanding directly informs the algorithmic solution - in this case, a dynamic programming approach where each state’s maximum profit is calculated based on the previous states.

The elegance and effectiveness of the code often directly correlate to how well the model captures the essential aspects of the problem. A well-constructed model will help to write more concise, efficient, and understandable code.

In conclusion, the art of problem-solving is significantly enhanced by the ability to create effective models. By simplifying complex problems into manageable models, you can better understand the problem and find elegant solutions. However, it’s always important to remember the limitations of your model and remain open to refining it as necessary.