• P is the class of problems with yes-or-no answers that can be solved by an algorithm whose runtime is polynomial in the number of bits of input. For example, one can test whether a list of points (given by Cartesian coordinates as binary numbers) contains any repeated points, by comparing each pair of points and checking whether they are equal. This algorithm takes time proportional to the square of its input length, a polynomial. It is also possible to solve the same problemmore quickly, using sorting or hashing.

• NP is a larger class of yes-or-no problems, not all of which are believed to have efficient algorithms.However, whenever the answer to a problem in NP is “yes,” there must exist a short and easily checked proof that the answer is yes.Many common puzzles, such as sudoku, can be formulated as problems in NP.1 For such puzzles, the yes-or-no question is “does this puzzle have a solution?” or “if I add this step tomy answer, will it still have a solution?” The proof that it does have a solution is the solution itself. Although the solution may be hard to find, it is easy to check that it is valid.