---
name: problem-approach
description: Write the `## Approach` section — a Mermaid flowchart that traces the submitted solution — for every docs solution page under apps/docs/content/ that lacks one, using the repo's exact placement and node conventions. Use when asked to add/implement/fill in missing approaches, draw the approach diagram for a problem page, or when `/problem-approach` is run.
---

# Problem approach diagrams

Every solution page (`apps/docs/content/(<phase>)/<topic>/(problems)/<num>_<slug>.mdx`) has a
hand-owned section before the script-owned one:

```
frontmatter → ::::warning (maybe) → ### Example N / ### Constraints
## Approach        ← this skill (Mermaid flowchart of the code below)
## Solution        ← bun run sync owns this; never edit inside it
```

`sync.ts` regenerates only `## Solution`, so an `## Approach` placed **before** it
survives every re-sync; anything placed inside `## Solution` is overwritten on the
next `work/` push.

## The command: fill every missing approach

1. **Find the pages** (numbered pages with no `## Approach` heading):

   ```sh
   grep -L '^## Approach' apps/docs/content/*/*.mdx
   ```

2. **For each page**, read `## Solution` and write the diagram from *that code* — the
   one on the page, not the textbook algorithm. The reader compares diagram and code
   line by line; the diagram is a trace, not a lecture.

3. **Insert** the section between the `### Constraints:` list and `## Solution`,
   blank-line separated on both sides:

   ````md
   - `-100 <= Node.val <= 100`

   ## Approach

   ```mermaid
   flowchart TD
     …
   ```

   ## Solution
   ````

4. **Check** each diagram by hand against the syntax rules below (the docs build does
   not parse Mermaid — a broken diagram fails only in the browser), then run the
   docs build once at the end and review the pages in `bun run dev`:

   ```sh
   cd apps/docs && bunx blume build --isolated && rm -rf .blume-verify
   ```

Pages with two languages (`<CodeGroup>`) get **one** diagram, traced from the
Python version. Pages with only JavaScript trace the JavaScript.

## Diagram conventions

Reference page: `apps/docs/content/(array)/15-3sum.mdx`. Every existing diagram
follows this shape:

```mermaid
flowchart TD
  S(["threeSum(nums)"]) --> O["nums.sort() — duplicates become adjacent"]
  O --> F{"more i in 0..n-1?"}
  F -- no --> E(["return result"])
  F -- yes --> D{"nums[i] == nums[i-1]?"}
  D -- yes --> F
  D -- no --> P["left = i+1, right = n-1, target = -nums[i]"]
  P --> W{"left < right?"}
  W -- no --> F
  W -- yes --> C["current = nums[left] + nums[right]"]
  C --> Q{"current vs target"}
  Q -- equal --> A["append triplet, skip equal neighbours, left += 1, right -= 1"]
  A --> W
  Q -- "current < target" --> L["left += 1 — need a bigger sum"]
  L --> W
  Q -- "current > target" --> R["right -= 1 — need a smaller sum"]
  R --> W
```

- `flowchart TD`, two-space indent, one edge per line, edges listed in execution
  order (entry first, exits as soon as they are reachable).
- **Entry** is a stadium holding the signature: `S(["threeSum(nums)"])`. **Every
  exit** is a stadium: `E(["return result"])`, `R(["return true"])`.
- **Statements** are rectangles `["…"]`, in the code's own identifiers and operators
  (`left += 1`, `freq[n] = (freq[n] or 0) + 1`). Several trivial assignments may share
  one node, comma-separated.
- **Decisions** are diamonds `{"…?"}` with `-- yes -->` / `-- no -->` edges. A
  three-way branch names the outcomes: `-- equal -->`, `-- "current < target" -->`
  (quote an edge label whenever it holds `<`, `>`, or spaces around operators).
- **Loops** are a decision: `for` → `{"more n in nums?"}`, `while` → the condition
  verbatim `{"left < right?"}`. `-- no -->` goes to whatever follows the loop; the
  body's last node edges back to the decision. `continue` is an edge back to the loop
  decision (`D -- yes --> F` above); `break` is an edge to the node after the loop.
- **Recursion** is a statement, not an expanded subgraph: `A["left = invertTree(root.left)"]`.
  The base case is the first decision after entry.
- **Classes** (design problems) use one `subgraph name["method(args)"]` per public
  method, plus a `state["…"]` subgraph describing the fields, with `state --> method`
  edges — see `hybrid-structures/(problems)/146_lru-cache.mdx`. Private helpers are named inside the
  method nodes (`"_remove_node then _add_node — move it beside the tail"`), not drawn.
- **Annotate the why with an em dash** inside the node, sparingly (one clause, a few
  per diagram): `"nums.sort() — duplicates become adjacent"`, `"map = {0: 1} — the
  empty prefix"`. The dash explains the invariant that makes the step correct — the
  one thing the code does not say.
- **Node ids**: one or two capitals, unique per diagram (`S I L Q C E R W A B`). Class
  diagrams prefix per method (`G1`, `P1`).

### Syntax rules (a diagram that breaks the page is worse than none)

- Wrap **every** label in double quotes — brackets, parentheses, `<`, `>` and `|` in an
  unquoted label break the parser.
- Never put `"` inside a label; use `'` for string literals (`"s == ''?"`).
- Never use `|` anywhere — not in node labels, not as `-->|text|` edge syntax; edge
  labels are always `-- text -->`. Write `or` for a logical or.
- Every id an edge references must be declared with a shape the first time it appears.
- Balanced: each `subgraph` has an `end`; each fence opens ```` ```mermaid ```` and closes ```` ``` ````.

### Code that does not run

Trace it anyway — the diagram documents what was submitted — and put a `::::warning`
callout **before** the fence naming exactly why it fails (`sliding-window/(problems)/121_best-time-to-buy-and-sell-stock.mdx`):

````md
## Approach

::::warning
The diagram traces the code below literally. That code does not run: `curr` and `ans` are never initialised, so the first `curr > left` test raises `NameError`.
::::

```mermaid
…
```
````

Do not fix the code on the page: `## Solution` mirrors `work/1`, and the fix belongs
there (then `bun run sync`).

## Do not

- Do not write prose approaches, complexity tables or alternative algorithms — the
  section is one flowchart (plus the warning callout when the code is broken).
- Do not touch `## Solution` or anything after it, nor the frontmatter `topics:` /
  `video:` keys (`bun run sync` / `bun run videos --write` own them).
- Do not add an approach to `reference/` pages or `index.mdx` — only numbered
  solution pages (`title: 'NN. …'`) carry one.