In mathematics, a **proof without words** (or **visual proof**) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature.^{[1]} When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.^{[2]}

A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.

The statement that the sum of all positive odd numbers up to 2*n* − 1 is a perfect square—more specifically, the perfect square *n*^{2}—can be demonstrated by a proof without words.^{[3]}

In one corner of a grid, a single block represents 1, the first square. That can be wrapped on two sides by a strip of three blocks (the next odd number) to make a 2 × 2 block: 4, the second square. Adding a further five blocks makes a 3 × 3 block: 9, the third square. This process can be continued indefinitely.

Main article: Pythagorean theorem § Rearrangement proofs |

The Pythagorean theorem that can be proven without words.^{[4]}

One method of doing so is to visualise a larger square of sides , with four right-angled triangles of sides , and in its corners, such that the space in the middle is a diagonal square with an area of . The four triangles can be rearranged within the larger square to split its unused space into two squares of and .^{[5]}

Jensen's inequality can also be proven graphically. A dashed curve along the *X* axis is the hypothetical distribution of *X*, while a dashed curve along the *Y* axis is the corresponding distribution of *Y* values. The convex mapping *Y*(*X*) increasingly "stretches" the distribution for increasing values of *X*.^{[6]}

*Mathematics Magazine* and the *College Mathematics Journal* run a regular feature titled "Proof without words" containing, as the title suggests, proofs without words.^{[3]} The Art of Problem Solving and USAMTS websites run Java applets illustrating proofs without words.^{[7]}^{[8]}

For a proof to be accepted by the mathematical community, it must logically show how the statement it aims to prove follows totally and inevitably from a set of assumptions.^{[9]} A proof without words might imply such an argument, but it does not make one directly, so it cannot take the place of a formal proof where one is required.^{[10]}^{[11]} Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.^{[12]}^{[13]}