Classic Signal: 'If n² Is P, Then n Is P' — Flip to Contrapositive

Certain shapes of theorem scream "contrapositive" the moment you read them. The most classic is this one:

If n^2 has property P, then n has property P.

If you can learn to spot this pattern in a single glance, you save yourself the wasted minutes of trying a doomed direct proof and you land on the contrapositive reflex immediately.

The shape to recognise

Read the claim and ask: "Does the hypothesis involve a higher power (square, cube, product, product of factors) of the variable, while the conclusion involves the variable itself at first power?"

If yes — direct proof is probably going to stall, and the contrapositive will almost certainly finish in one line.

Examples of the shape:

"If n^2 is even, then n is even."
"If n^2 is odd, then n is odd."
"If n^3 is divisible by 5, then n is divisible by 5."
"If ab is odd, then a is odd and b is odd."
"If x^2 is rational, is x rational?" (The answer is no in general — but whenever a claim of this shape is true, the contrapositive is the way to prove it.)

In each case the hypothesis carries the product or power — that is the unwieldy thing — and the conclusion is about the unpowered variable.

Why direct proof stalls

A direct proof of "if n^2 is P then n is P" asks you to start with an expression for n^2 and extract information about n. But the step from n^2 to n is a square root — not an algebraic operation that produces clean forms.

Take "if n^2 is even, then n is even." Start with n^2 = 2k. You want n = 2m for some integer m. Squaring n = 2m would give n^2 = 4m^2, which is a stronger condition than n^2 = 2k — so you cannot simply invert. And taking \sqrt{2k} does not expose a factor of 2 in n; it just says n squares to an even number, which is where you already are.

Why un-squaring fails: the equation n^2 = 2k does not pin down n tightly enough. Many integers n have squares of the form 2k, and knowing this form alone is not enough to say whether n itself is a multiple of 2. The direct proof would need exactly the theorem you are trying to prove.

Why the contrapositive finishes instantly

The contrapositive of "if n^2 is P, then n is P" is "if n is not P, then n^2 is not P." Now the hypothesis is about the unpowered variable (n), and the conclusion is about the power (n^2). The direction is reversed — and reversed in exactly the favourable way: you start with a clean expression for n and you are allowed to square it, which is an easy algebraic operation.

Take "if n^2 is even, then n is even" again. Contrapositive: "if n is odd, then n^2 is odd." Start with n = 2k + 1. Square: n^2 = (2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1, which is odd. One line. Done.

The asymmetry between un-squaring (hard) and squaring (easy) is the whole reason this pattern favours the contrapositive.

The recognition reflex

Build the reflex with one question:

Is the hypothesis a power of the variable (or a product of variables) while the conclusion is about the variable itself?

If yes, your first move is contrapositive — do not even try direct. Write the contrapositive on the left side of the page, and prove that instead. This saves you the detour through a direct proof that will run aground at the un-squaring step.

Five claims with the same shape: a power (or product, or sum) on the hypothesis side, a simple variable on the conclusion side. In every case the direct route tries to invert the operation — square root, cube root, product split — and stalls. The contrapositive lets you apply the operation in the forward direction, which is always cleaner.

The micro-checklist

When you see a claim and want to decide in five seconds whether to flip:

Is the hypothesis a power, product, or composite expression of the variable? Yes → flip signal.
Is the conclusion a simple (first-power) statement about the variable? Yes → flip signal.
Would the direct proof need to invert the hypothesis operation (un-square, un-cube, un-multiply)? Yes → flip signal.

If all three check out, do not attempt the direct proof. Write "Contrapositive: if n is not P, then n^2 is not P" at the top of the page and proceed.

Extending the pattern beyond squaring

The pattern generalises cleanly to any operation where the forward direction is easy and the inverse is hard:

Cubing. "If n^3 is divisible by 2, then n is divisible by 2." Un-cubing is unwieldy; cubing is a single-line identity. Flip.
Multiplying. "If ab is even, then a is even or b is even." Splitting a product is hard; multiplying is easy. Flip.
Summing. "If a + b is rational, and a is irrational, then b is irrational." Un-summing with mixed rationality is awkward; adding a rational and an irrational is easy. Flip — the contrapositive says "if b is rational, then a + b is rational or a is rational," which is a one-line algebra fact.
Composing functions. "If g \circ f is injective, then f is injective." Un-composing is hard; composing is straightforward. The contrapositive "if f is not injective, then g \circ f is not injective" follows by picking two distinct inputs of f that collapse.

Every time the hypothesis contains an operation whose inverse is nasty, flip.

The rule in one sentence

If the hypothesis is a higher-power (or product, or composed) expression of the variable and the conclusion is about the variable itself at first power, reach for the contrapositive. That is the single most productive pattern recognition in the entire contrapositive chapter, and spotting it on sight is the main skill to build.