External Exercises

This module should be practiced locally first. Start with the concept drills, then use these exercise lanes to deepen speed, flexibility, and proof quality.

Local Book Exercise Lanes

Counting and Combinatorial Proofs

• MCS: Cardinality Rules Problems and References (Part 1) - product rule, division rule, subsets, and early counting setups
• MCS: Cardinality Rules Problems and References (Part 2) - more structured counting and case analysis
• MCS: Cardinality Rules Problems and References (Part 3) - inclusion-exclusion and counting proof practice
• Rosen: Chapter 6 Review / Supplementary Exercises - extra exercise volume after core mastery

Recurrences and Generating Functions

• MCS: Generating Functions Problems and References (Part 1) - coefficient extraction and model translation
• MCS: Generating Functions Problems and References (Part 2) - deeper recurrence and generating-function exercises
• Rosen: Applications of Recurrence Relations (Part 4) - recurrence construction and interpretation

Graph Structure and Trees

• MCS: Simple Graphs Problems and References (Part 1) - degrees, paths, connectivity
• MCS: Simple Graphs Problems and References (Part 2) - bipartite graphs, coloring, and matchings
• MCS: Simple Graphs Problems and References (Part 3) - tree reasoning and special graph families
• Rosen: Chapter 11 Review / Supplementary Exercises - tree and spanning-tree extension work

Self-Curated Proof Set

Build a custom problem set with these minimums:

• 5 counting problems where you justify the setup before computing
• 3 combinatorial identity proofs
• 3 inclusion-exclusion or pigeonhole arguments
• 3 recurrence-modeling problems
• 5 graph proofs using degree sums, path arguments, contradiction, or induction
• 2 planarity, coloring, or matching problems

Completion Checklist

• Completed at least 10 local counting exercises with written setup justification
• Completed at least 5 local graph exercises with proof-quality explanations
• Solved at least 3 recurrence or generating-function exercises
• Wrote at least 3 combinatorial proofs, not just numeric solutions
• Logged mistakes and corrected setups for at least 5 missed problems

Progress Tracking

Exercise type	Goal	Done
Counting setup and calculation	10	___
Constrained counting proofs	6	___
Recurrence and generating functions	5	___
Core graph structure proofs	8	___
Trees, coloring, or planarity	5	___
Total	34	___

Use Strategy

• Do not do 20 problems of the same surface form in a row.
• Mix one clean computation problem with one proof problem and one translation problem.
• After every missed exercise, write one line naming the modeling error: ordered/unordered, double-counted overlap, wrong graph model, bad recurrence state, or used theorem without hypotheses.