docs(docs): update problem page paths to use (problems)/<num>_<slug>.mdx format and adjust related examples

.claude/skills/problem-approach/SKILL.md

@@ -5,7 +5,7 @@ description: Write the `## Approach` section — a Mermaid flowchart that traces
 
 # Problem approach diagrams
 
-Every solution page (`apps/docs/content/(<phase>)/<topic>/<num>-<slug>.mdx`) has a
+Every solution page (`apps/docs/content/(<phase>)/<topic>/(problems)/<num>_<slug>.mdx`) has a
 hand-owned section before the script-owned one:
 
 ```
@@ -100,7 +100,7 @@ flowchart TD
   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 `(hash-table)/146-lru-cache.mdx`. Private helpers are named inside the
+  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
@@ -122,7 +122,7 @@ flowchart TD
 ### 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 (`(array)/121-best-time-to-buy-and-sell-stock.mdx`):
+callout **before** the fence naming exactly why it fails (`sliding-window/(problems)/121_best-time-to-buy-and-sell-stock.mdx`):
 
 ````md
 ## Approach