Sparkle

Everyone says "think from first principles" in competitive programming, but very few explain what it actually looks like when you're stuck on a problem. It's not any trick or formula. It's a way of thinking. It means you stop asking "which algorithm fits here?" and instead ask "what is actually happening here?' Let's understand this with a simple example. Problem: You're given an array. In one operation, you can choose any element and decrease it by 1. Your goal is to make all elements equal using minimum operations. > Most people instantly jump to ideas like sorting, greedy templates, or some known pattern. But first principles thinking starts differently. 1. You ignore everything and just observe. 2. Take a small example: [3, 1, 2] 3. Now simulate manually. To make all elements equal, you can only decrease values. So clearly, everything has to come down to the smallest element. Why? Because you can't increase anything. So our target is minimum element. Now count: 3 -> 1 takes 2 operations 2 -> 1 takes 1 operation Total = 3 operations. You didn't use any fancy algorithm. You just asked: 1. What operations are allowed? 2. What is even possible? That's first principles. To think fundamentally Let's go one step deeper. Consider another problem: You're given a binary string. In one move, you can delete any adjacent pair "01". What's the maximum number of deletions? Now instead of guessing "stack?', "greedy?", pause. Try small cases: "01" -> 1 "0011" -> try manually Remove middle -> "01" -> 1 more -> total 2 Now observe: Every deletion removes one '0' and one '1'. So the total deletions can never exceed: min(count of 0s, count of 1s) That:s it. You didn't derive this from memory. You discovered it. This is the real difference. Average approach: 'Which topic is this? Have I seen this before?" First principles approach: "What must always be true here?" Another important habit is asking extreme questions: 1. What if n = 1? 2. What if all values are same? 3. What if operations are applied infinitely? These questions expose constraints and invariants faster than any template. Also, Please for the God's sake don't rush coding. The biggest mistake in contests is forcing a known approach too early. You see something slightly similar to a DP problem -> you start writing DP. You see a graph -> you start BFS/DFS. But first principles thinking says: "Understand first. Code later." Spend time breaking the problem. Write examples. Find patterns. Only then translate into code. At the end of the day, first principles thinking is simple: - Strip the problem. - Understand the rules. - Observe small cases. - Find what must always hold. - Build the solution from zero. Do this consistently, and you'll notice something: You will stop fearing new problems. Because no matter how unfamiliar they look, you know how to start. Reposts are appreciated ♥️♥️