Wishful Thinking and Make it Easier
These strategies combine psychology and mathematics to break initial impasses during your practice session. Ask yourself:
- What is it about the problem that makes it hard?
Then, make the difficulty disappear. You may not be able to do this legally, but who cares? Temporarily avoiding the hard part of a problem will allow you to make progress and may shed light on the difficulties. For example, if the problem involves big numbers, make them small.
If a problem involves complicated constraints, try looking at a similar problem without such constraints. At best, pretending that the difficulty isn’t there will lead to a bold solution. At worst, you will be forced to focus on the key difficulty of your problem and possibly formulate an intermediate question, whose answer will help you with the problem at hand.
These strategies can be used to make some initial progress. It is important to keep in mind that any progress is ok. Never be in a hurry to solve a problem! The process of investigation is just as important. Importance of Thinking about the Problem
If I had an hour to solve a problem, I’d spend 55 minutes thinking about the problem and 5 minutes thinking about solutions. Albert Einstein
Time spent thinking about a problem is always time worth spent. Even if you seem to make no progress at all. Experiment with different ideas
Making the Problem Approachable
The principle of “Wishful Thinking” and “Make it Easier” revolves around the idea of simplifying a problem to make it more approachable. The core concept is to temporarily ignore or simplify the aspects that make the problem complex or challenging.
For instance, suppose you’re tasked with creating an algorithm that handles a large dataset with numerous constraints. The sheer size of the data and the constraints could be overwhelming. However, using the “Wishful Thinking” or “Make it Easier” strategy, you might choose to ignore the size and constraints for a while and focus on developing a solution for a smaller subset of data.
Example: Consider a problem where you need to design an algorithm to find the shortest path between two nodes in a large, weighted, and directed graph. This is a complex problem due to the size and the nature of the graph (weighted and directed).
However, if you apply the “Make it Easier” strategy, you could start by considering a smaller, unweighted, and undirected graph. This way, you might be able to develop an initial solution more quickly, which can then be expanded and modified to handle the complexities of the original problem.
Importance of Thinking about the Problem
Spending time understanding and simplifying a problem is a crucial step in problem-solving. It allows you to see the problem more clearly and identify potential solutions more effectively.
Albert Einstein’s quote emphasizes the importance of spending time understanding the problem before jumping to the solution. This understanding phase includes identifying the problem’s key elements, constraints, and potential hurdles. Even if it feels like you’re not making direct progress towards a solution, this time spent will often lead to more efficient and effective solutions in the end.
Experiment with Different Ideas
When solving a problem, it’s essential to remain open and flexible, allowing yourself to try different approaches. This experimentation can often lead to unexpected insights or solutions that might not have been evident initially.
For example, when working on an algorithmic problem, you might start with a brute-force solution. Although this might not be the most efficient solution, it provides a starting point from which you can refine and optimize. As you iterate on your initial solution, you can experiment with different algorithms, data structures, and optimizations, which can help you discover a more effective and efficient solution.
Summary
Here is a summary of the key points:
Wishful thinking and make it easier are heuristics to tackle difficult problems by temporarily simplifying or ignoring the hard parts.
For example, faced with a problem involving large datasets or constraints, try developing a solution for a smaller subset without those constraints.
Spending time understanding the problem before solving is crucial. Experiment with different approaches, even if they don’t seem to make progress initially.
Albert Einstein emphasized thinking about the problem for 55 minutes out of an hour, rather than rushing to solutions.
Making the problem easier allows you to make some initial progress. This can provide insights into the difficulties and help formulate intermediate sub-problems.
The process of investigation is as important as finding the solution. Simplifying the problem makes it more approachable but the key difficulties still need to be addressed eventually.
These strategies combine psychology and mathematics to tackle impasses. Make bold temporary simplifications to gain insights into productive ways forward.
The key ideas are to buy time by simplifying, understand the core difficulties, and iterate with different bold ideas through experimentation. Even if progress seems slow, the investigation process will lead to effective solutions.