• i. Attacking a Problem: A working toolkit for hard problems:
• ii. Invariants, Parity and Extremal Arguments: The three arguments behind most ad hoc and constructive problems:
• iii. Contribution and Complementary Counting: The two most common moves in counting problems:
• iv. Proof Techniques: Proof is what turns “passes the samples” into “is correct”.
• v. Constructive Algorithms: Constructive problems ask you to build an object, or report that none exists.
• vi. Classic Puzzles: Famous puzzles that keep showing up as contest problems or as the core of one.
• vii. Observation Toolkit: the brute force $\to$ table $\to$ guess $\to$ prove loop and the moves that produce observations (invariants, extremes, reversing the process, canonical forms).
• viii. When the Answer Is Small: answers bounded by a constant: universal strategy, exact characterization of the cheaper cases, lower-bound proofs (permutation sort, mex to zero).
• ix. Sorting and Ordering Observations: inversions and cycles, relative-order invariants, ants on a stick, sorted-order patterns (median, pairing, prefix maxima).
• x. Parity, Coloring and Mod Observations: invariants mod k, permutation parity, the 15-puzzle solvability test, colouring arguments for tilings.
• xi. Operations and Reachable States: reachability by canonical forms and invariants, swaps at distance k, commuting and self-inverse operations.
• xii. Ad Hoc on Arrays and Strings: binary strings, alternating targets, palindrome rearrangement, MEX and run observations.
• xiii. Ad Hoc Problem Set: practice problems (CSES, Codeforces, classics) with the key observation for each.
• xiv. Permutation Constructions: LIS/LDS (Erdős–Szekeres), exactly k inversions, no adjacent consecutive values, derangements, involutions.
• xv. Graph Constructions: Havel–Hakimi, trees with given diameter or degrees, k shortest paths, Mycielski, Turán, regular graphs.
• xvi. Grid and Matrix Constructions: Latin and magic squares (odd, doubly even, LUX), snake paths, matrices with given row and column sums.
• xvii. Sequence and Number Constructions: 3-AP-free and Sidon sets, binary splitting, gcd tricks, xor-and-sum pairs, sum-free sets.
• xviii. Invariant-Based Constructions: invariant gives the condition, construction proves sufficiency; equalize by triple xor.
• xix. Greedy Constructions: lexicographically smallest objects with a feasibility test; I/D pattern permutations.
• xx. Interactive and Output-Only Constructions: constructing by queries (hidden permutation with mod queries), adaptive judges, output-only search.
• xxi. Problem Reduction and Modeling: Most hard problems are a known problem in disguise.