You are staring at a claim "if P then Q" and the forward path from P is not obvious. You try a different angle: what would Q follow from? If I knew R, could I get Q? And what would I need to get R? A few minutes later you have a chain running from Q back to something that looks a lot like P. A small voice asks: isn't that cheating?

The short answer: no. Working backwards from the conclusion is how most proofs are actually discovered. What would be cheating is writing the proof down that way. The distinction between finding a proof and presenting one is the key.

Two different jobs: exploration vs presentation

Every proof goes through two phases, and the rules for each are different.

Phase 1 — exploration (private). You are hunting for a path from P to Q. All methods are fair game: try small cases, guess the algebraic form, run the argument forward and backward, sketch pictures, ask "what would make Q true?" Whatever gets you to see the path.

Phase 2 — presentation (public). You write the final proof. Here the rules are strict: the chain must run from P (or another firmly-established starting point) to Q, with every step justified. The reader should not need to know how you found the proof — only that it is correct.

Working backwards is a Phase-1 activity. It is a tool for discovering the connection. The written proof, however, belongs to Phase 2 — it must run forwards. Confusing the two phases is what causes trouble.

Why the two-phase rule matters: a proof that reads backwards — "we want to show Q; it would suffice to show R; and R would follow from P; therefore Q" — is often logically okay but carries a hidden risk. At each backward step you have to check that the implication actually runs both ways. If Q follows from R, but R does not follow from anything you can establish, the backward chain is broken and the reader cannot tell where. Running the chain forwards forces you to verify every implication in the direction it actually needs to work.

Why working backwards is honest

Working backwards does not add anything to the proof that was not logically there. You are asking: for which statements R does R \Rightarrow Q hold? Each candidate R is a potential stepping-stone. If one of those candidates can itself be derived from P, you have found a bridge — P \Rightarrow R \Rightarrow Q — and the direction you discovered it in does not matter. The logical chain is the same chain either way.

A useful picture: the proof is a path through a graph whose nodes are statements and whose edges are legal implications. Exploring backwards from Q and exploring forwards from P are two different search strategies for the same path. Once you find it, the path is the path.

The flip: from discovery order to presentation order

Discovering a proof backwards, then writing it forwardsTwo diagrams showing the same four-node chain from hypothesis P through intermediate steps R1 and R2 to conclusion Q. The first diagram shows the discovery order, with arrows going right to left from Q back to P as the student works out what would imply Q. The second diagram shows the presentation order, with arrows going left to right from P through R1 and R2 to Q. A draggable dot switches between the two views. P R₁ R₂ Q The same four nodes. Only the direction of search changes. drag to flip between the two search directions
The path from $P$ to $Q$ is the same path regardless of how you found it. Working backwards (red) is a fine way to discover the nodes $R_1, R_2$ that connect the two ends. Writing forwards (green) is what the final proof has to look like. The flip happens only in the notebook — the reader sees the forward version.

A concrete example

Claim. Prove that for every real x, (x + 1)^2 \ge 4x.

Phase 1 — exploration (working backwards). I want (x+1)^2 \ge 4x. Rearrange: (x+1)^2 - 4x \ge 0. Expand: x^2 + 2x + 1 - 4x \ge 0, i.e. x^2 - 2x + 1 \ge 0, i.e. (x-1)^2 \ge 0. That is true for every real x, since squares are never negative. So the claim follows if I can run this chain the other way.

Phase 2 — writing forwards. For any real x, (x - 1)^2 \ge 0 because a real square is never negative. Expanding, x^2 - 2x + 1 \ge 0. Adding 4x to both sides, x^2 + 2x + 1 \ge 4x, i.e. (x + 1)^2 \ge 4x. \square

The written proof runs left-to-right, every step justified. But the discovery went right-to-left: start at the goal, rearrange until you hit something you know is true, then reverse the chain.

The one honest worry: reversibility

Backward reasoning is safe only when each step is reversible — each implication in your backward chain must also hold in the forward direction. Otherwise the backward chain finds you a "proof" that runs forward only halfway.

The classic trap is squaring both sides of an equation. If you want to prove \sqrt{x+1} = x - 1 for a particular x, working backwards you might write:

Squaring both sides: x + 1 = (x - 1)^2 = x^2 - 2x + 1, so x^2 - 3x = 0, so x = 0 or x = 3.

Now read it forwards: "if x = 3, then squaring both sides gives \sqrt{4} = 2 and 3 - 1 = 2 — they match." Fine for x = 3. But: "if x = 0, then \sqrt{1} = 1 and 0 - 1 = -1 — they do not match." The backward chain introduced a solution that does not satisfy the original equation, because squaring is not reversible (a = b implies a^2 = b^2, but a^2 = b^2 does not imply a = b).

Fix: every step in a backward derivation must be an iff ("if and only if") — a two-way implication. The (x+1)^2 \ge 4x example above worked because adding a constant, expanding, and factoring are all reversible operations. The squaring example failed because squaring is one-way.

Why reversibility is the hinge: a backward discovery plus reversibility at every step is equivalent to a forward proof. Without reversibility, the backward chain may pass through statements strictly weaker than what you need, and the forward version will be broken at the non-reversible step. Always write "iff" instead of "⇒" during backward exploration — it forces you to notice when a step is one-way.

The practical workflow

A reliable way to use backward reasoning without falling into the reversibility trap:

  1. On scratch paper, work from Q backwards towards P, using "iff" at each step.
  2. If a step is not an iff (one-way only), mark it and stop — you need a different backward step, or a forward argument at that link.
  3. Once the chain reaches something obviously implied by P, you have the route.
  4. Rewrite the chain in the opposite direction. Each step becomes a because-assertion (see Because vs If).
  5. Check every step as a forward implication — do not rely on having checked them backwards.
  6. Hand in the forward version only.

This is exactly what mathematicians do. The public face of mathematics is the clean forward proof; the private workshop is full of backward reasoning, scratch work, and guessed-then-verified identities.

The short summary

The only thing that would be cheating is writing down a proof that looks like it runs forwards but was not checked to run forwards — where "works backwards" and "works forwards" diverge at some hidden one-way step. Do the flip honestly, and backward reasoning is one of the most powerful tools you have.

Related: Mathematical Proof — Direct Proof · Because vs If Inside a Proof · Scratch Work vs Final Proof · What You Can Assume When You Assume P