The Big Four Observations

Most "aha" moments in HSG problems reduce to one of four families.

1. Invariants

A quantity that never changes under the allowed operation. If you can name it, every reachable state must preserve it — and unreachable goals die instantly. Ask: what is conserved? (sum, XOR, sum of signs, count of inversions mod 2, ...)

2. Parity

Many quantities only matter mod 2: alternation, pairing, coloring. A parity argument can collapse an exponential search into a one-line check. Ask: does only even/odd matter here?

3. Monotonicity

If more of X can only help (or only hurt), binary search over X becomes legal and greedy sorting becomes safe. Ask: does the objective move in one direction along some axis?

4. Symmetry

Identical objects, mirror structures, exchangeability of two choices. Symmetry halves work, deduplicates states, and powers exchange arguments for greedy proofs. Ask: what can I swap without changing the answer?

Applying them

Run the experiment loop first. When the table is in front of you, test each family against it: does a column never change (invariant)? Does the answer depend on n mod 2 (parity)? Does it move one way as the input grows (monotonicity)? Do two rows always agree (symmetry)? Then prove.