Proof without words

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.

[1] When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.

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 2n − 1 is a perfect square—more specifically, the perfect square n2—can be demonstrated by a proof without words.

[3] In one corner of a grid, a single block represents 1, the first square.

[4] One method of doing so is to visualise a larger square of sides

in its corners, such that the space in the middle is a diagonal square with an area of

[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.

[10][11] Rather, mathematicians use proofs without words as illustrations and teaching aids for ideas that have already been proven formally.

Proof without words of the Nicomachus theorem ( Gulley (2010) ) that the sum of the first n cubes is the square of the n th triangular number
A proof without words for the sum of odd numbers theorem
Rearrangement proof of the Pythagorean theorem. The uncovered area of gray space remains constant before and after the rearrangement of the triangles: on the left it is shown to equal c² , and on the right a²+b² .
A graphical proof of Jensen's inequality