The Unchanging Truth

Intro to Proofs · Axiom Academy

Some puzzles hide a quantity that never changes, no matter what moves you make. Find it, and you can prove the impossible without trying forever. A puzzle gives you a starting state and a few legal moves, and asks: can you reach some goal? Trying move after move can go on forever. The invariant method flips the question — instead of chasing the goal, you hunt for a quantity that every legal move leaves unchanged . If the goal has a different value of that quantity, it is unreachable, and you have a proof. Here is the simplest version. The token starts at 7 , and the only moves allowed are to add or subtract an even number . Watch the value change at every step — and watch the one thing that never does. An even move can shift the value by any amount — but it can never flip odd to even. That fixed parity is the invariant. Some goals are simply out of reach Drag the token. Each even move slides it, but notice it can only ever land on the odd marks — it can never sit on an even one. So try to park it on the green target ( 15 , odd) and then on the red target ( 12 , even). One you can hit. The other the invariant forbids. No sequence of even moves can change an odd number into an even one — so 12 stays unreachable forever, and you never have to test a single path to know it. The board that can't be covered

This is the written version of the interactive lesson above. See the full Intro to Proofs course.