Add the problem index and 22 problem walkthroughs

[design-gurus]

problems/README.md maps all 302 problems in the course to the pattern that solves them, with the difficulty the course assigns. It is generated from the live course tree, so every count on the page is computed rather than typed, and it links each pattern page.

22 problems get a walkthrough, covering 20 of the 41 patterns. Each one gives the problem in our own words, the argument for why it belongs to its pattern, the approach and its invariant, the complexity, and the edge cases to say out loud. They stop short of the code, because the worked solutions are what the course is for.

Two things the index makes explicit that the course does not:

• Three problems are genuinely solvable under two patterns (Number of Provinces, Sum of Subarray Minimums, Next Greater Element).
• Balanced Parentheses is two different problems with one name: validating a string under Stacks, and generating every valid string under Subsets.

Root README gains a problem index section, and the coming-next list is updated.