Proof without words
In mathematics, a proof without words 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. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.
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.
Examples
Sum of odd numbers
[Image:Proofwithoutwords.svg|thumb|upright|A proof without words for the sum of odd numbers theorem]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.
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 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.
Pythagorean theorem
The Pythagorean theorem that can be proven without words.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.