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+# 1365. How Many Numbers Are Smaller Than the Current Number
+
+**Difficulty:** Easy
+**URL:** https://leetcode.com/problems/how-many-numbers-are-smaller-than-the-current-number/
+**Topics:** Array, Hash Table, Sorting, Counting Sort
+
+---
+
+## The One Insight That Makes This Work
+
+> If I know how many times each value appears, and I add those counts up from left to right, then `prefix[v]` tells me **how many elements are ≤ v** — instantly, for any v.
+
+That's it. Everything below is just executing this idea.
+
+---
+
+## Step 1: Spot the Signal
+
+Read the constraints:
+
+```
+0 <= nums[i] <= 100
+```
+
+Values are bounded to a tiny range (0–100). This is the **flashing neon sign** that says: don't sort, don't nest loops — build a frequency array indexed by value.
+
+**Rule of thumb:** value range ≤ ~10⁶ and you need counting/ranking? Frequency array.
+
+---
+
+## Step 2: Count Every Value (the "bucket" pass)
+
+Make an array with one slot per possible value. Walk the input once. Each number votes for its own slot.
+
+```javascript
+const freq = new Array(101).fill(0);  // slots for values 0..100
+for (const x of nums) freq[x]++;
+```
+
+For `nums = [8, 1, 2, 2, 3]`:
+
+```
+value:  0  1  2  3  4  5  6  7  8  ...
+freq:   0  1  2  1  0  0  0  0  1  ...
+```
+
+Read it as: "one 1, two 2s, one 3, one 8."
+
+---
+
+## Step 3: Prefix Sum (the magic pass)
+
+Now transform `freq` in place: each slot becomes itself **plus everything before it**.
+
+```javascript
+for (let i = 1; i < 101; i++) freq[i] += freq[i - 1];
+```
+
+Same example after the pass:
+
+```
+value:  0  1  2  3  4  5  6  7  8  ...
+freq:   0  1  3  4  4  4  4  4  5  ...
+```
+
+New meaning: `freq[v]` = **count of elements ≤ v**.
+
+- `freq[3] = 4` → four numbers are ≤ 3 (they are 1, 2, 2, 3) ✓
+- `freq[7] = 4` → still four numbers ≤ 7 ✓
+
+---
+
+## Step 4: Answer Queries in O(1)
+
+"How many numbers are **strictly smaller** than x?" is the same question as "how many numbers are **≤ x − 1**?"
+
+```javascript
+return nums.map((x) => (x === 0 ? 0 : freq[x - 1]));
+```
+
+The `x === 0` guard exists because nothing can be smaller than the minimum possible value — and `freq[-1]` would be `undefined`.
+
+Trace on `[8, 1, 2, 2, 3]`:
+
+| x | lookup | answer |
+|---|--------|--------|
+| 8 | freq[7] | 4 |
+| 1 | freq[0] | 0 |
+| 2 | freq[1] | 1 |
+| 2 | freq[1] | 1 |
+| 3 | freq[2] | 3 |
+
+→ `[4, 0, 1, 1, 3]` ✓
+
+---
+
+## Full Solution
+
+```javascript
+var smallerNumbersThanCurrent = function (nums) {
+  // 1. Bucket counts
+  const freq = new Array(101).fill(0);
+  for (const x of nums) freq[x]++;
+
+  // 2. Prefix sum: freq[v] = count of elements <= v
+  for (let i = 1; i < 101; i++) freq[i] += freq[i - 1];
+
+  // 3. Strictly smaller than x == count of elements <= x-1
+  return nums.map((x) => (x === 0 ? 0 : freq[x - 1]));
+};
+```
+
+**Complexity:** O(n + k) time, O(k) space, where k = value range (101 here). No sort, no log factor.
+
+---
+
+## The Reusable Pattern (memorize this shape)
+
+```
+1. BUCKET   — freq[value]++ for every element
+2. PREFIX   — freq[i] += freq[i-1] left to right
+3. QUERY    — freq[v] answers "how many ≤ v" in O(1)
+              freq[v-1] answers "how many < v"
+              n - freq[v] answers "how many > v"
+```
+
+### Where else this exact shape shows up
+
+| Problem | Same pattern, different query |
+|---|---|
+| **Counting Sort** | Prefix sums become final sorted positions |
+| **LC 315 / rank queries** | "How many smaller" is literally a rank |
+| **LC 1122 Relative Sort Array** | Bucket + walk buckets in order |
+| **Radix sort digit pass** | Bucket by digit, prefix for placement |
+| **Histogram percentiles** | freq[v] / n = percentile of v |
+| **"How many in range [a, b]?"** | freq[b] − freq[a−1] — the prefix subtraction trick |
+
+### The generalization ladder
+
+- Values bounded and small → **frequency array** (this pattern)
+- Values huge but few distinct → **coordinate compression** first, then this pattern
+- Need updates between queries → upgrade prefix array to a **Fenwick tree (BIT)** — same idea, log-time updates
+
+---
+
+## Common Mistakes
+
+1. **Returning `freq[x]` instead of `freq[x-1]`** — that counts elements ≤ x (including x itself and its duplicates). Off-by-one between "≤" and "<" is where this pattern bites.
+2. **Forgetting the `x === 0` edge** — smallest possible value has nothing below it.
+3. **Sizing the array to `nums.length` instead of the value range** — the buckets are indexed by *value*, not position.