100 Examples Checked vs One Direct Proof: Why Examples Never Finish the Job

You have checked "even + even = even" on 2 + 4, on 6 + 10, on 100 + 48, and on every pair of even numbers your pencil reaches. After a hundred examples, the pattern seems impossible to break. So why is the hundred-example stack not a proof? Because the statement is "for every pair of even integers," and there are infinitely many pairs. Any finite check leaves infinitely many untested — and nothing in an example-stack rules out a counterexample at case 10^{100} + 1.

A direct proof, by contrast, handles the infinite in a single stroke. Five lines. One algebraic rearrangement. Every even pair, dealt with at once. The side-by-side below shows the two approaches running in parallel — one ticking up example by example, the other standing still because it has already finished.

The side-by-side reel

Speed:

Left panel: the counter advances and checks a new even pair $(a,b)$ each tick. The cases remaining readout stays at $\infty$ no matter how long you let it run. Right panel: the five-line direct proof closes the claim for every pair in one pass.

The five-line proof, unpacked

The proof on the right of the reel uses the definition of even as a tool, not a label.

Definition. An integer n is even if there exists an integer k such that n = 2k.

With that in hand, the argument is mechanical.

Suppose a and b are even. By definition, a = 2k for some integer k, and b = 2m for some integer m.
Add them. a + b = 2k + 2m.
Factor. 2k + 2m = 2(k + m).
Observe. k + m is an integer (integers are closed under addition).
Conclude. a + b = 2 \cdot (\text{integer}), so a + b is even by the definition. \blacksquare

Why this proof handles every pair at once: lines 1–5 do not mention any specific numbers. The letters a, b, k, m stand for arbitrary unknowns satisfying the hypothesis. The reasoning is valid no matter what specific values they take — so the conclusion is valid for every choice. That is what "for all" means: one argument, universally applicable.

Why the example-stack fails

Three reasons, each enough on its own.

Finite count vs infinite claim. The integers are infinite, so any finite check misses infinitely many cases.
Pattern matching is not proof. Many patterns hold for the first 10^9 cases and fail at the 10^9 + 1-th. The conjecture "n^2 + n + 41 is prime for all n" holds for n = 0, 1, 2, \ldots, 39 and then breaks at n = 40.
Examples confirm, they do not rule out. An example that fits the pattern does not rule out a future example that does not.

A direct proof eliminates all three problems. It argues about the form of the objects — "any integer of the form 2k" — so the argument does not care which specific integers you plug in.

Examples are still useful (just not as proof)

Do not walk away thinking examples are pointless. They serve three functions that proofs alone cannot:

Building intuition. Checking 2 + 4 = 6 tells you the claim is plausible. Without that plausibility check, you might try to prove a false statement.
Finding counterexamples. If a claim is false, a single example that breaks it is a complete disproof — no proof needed.
Discovering the proof. Working 2 + 4 = 6 as 2(1) + 2(2) = 2(1 + 2) = 2 \cdot 3 is the experiment that reveals the factoring pattern. The general proof is the same algebra, written with letters.

The rule is simple: examples suggest, proofs certify. Use both.

A second side-by-side — a trickier claim

Claim. The square of any odd integer is odd.

Example-stack attempt: 3^2 = 9 (odd ✓), 5^2 = 25 (odd ✓), 7^2 = 49 (odd ✓), 11^2 = 121 (odd ✓). A hundred more pass. Still nothing proven — just a hundred confirmations in an infinite sea.

Direct proof: let n = 2k + 1 be an arbitrary odd integer. Then

n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1.

The expression 2k^2 + 2k is an integer, so n^2 has the form 2(\text{integer}) + 1 — it is odd. \blacksquare

Five lines. Every odd integer in the universe, dealt with.

The ethos

Mathematics demands certainty, not confidence. A direct proof converts an infinite claim into a finite argument by reasoning about structure — the form of the objects — rather than their specific values. Examples live in the realm of finite checks; proofs live in the realm of structural arguments. Knowing when to reach for each is the first lesson of proof-writing.