The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.
It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.
The four color theorem was the first major theorem to be proven using a computer, and the proof is disputed by some mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.
The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!" (Full article...)
An animated geometric proof
of the Pythagorean theorem
, which states that among the three sides of a right triangle
, the square of the hypotenuse
is equal to the sum of the squares of the other two sides, written as a2 + b2 = c2.
A large square is formed with area c2
, from four identical right triangles with sides a
, fitted around a small central square (of side length b − a
). Then two rectangles are formed with sides a
by moving the triangles. Combining the smaller square with these rectangles produces two squares of areas a2
, which together must have the same area as the initial large square. This is a somewhat subtle example of a proof without words