Cryptarithmetic problems are mathematical problems in which each digit is replaced by a letter. Solving the problem consists of finding which letter is associated with each digit. If you have the right substitution, the arithmetic problem will be correct. For example, a very famous problem is: SEND + MORE = MONEY. Like all good cryptarithmetic problems, the letters make words that form a meaningful phrase. Solving these problems proceeds one step at a time. With a little examination, M = 1 because M is a "carry." Replacing all the Ms with 1s will get: SEND + 1ORE = 1ONEY. More reflection reveals each substitution in turn until you finally have: 9,567 + 1,085 = 10,652.
Sudoku is a Japanese puzzle form that has become the obsession of millions of people all around the world. It consists of a 9 x 9 set of squares, each just big enough to hold one digit. The 9 x 9 matrix is further divided into 3 x 3 matrices. When the puzzle is solved, the digits 1 through 9 will appear once each in every row, every column and every position of the 3 x 3 sub-matrices. Although it might seem surprising that one such arrangement could be found, there are actually billions. The puzzle starts with some of the squares filled in.
Rubik's cube was invented by an architect named Erno Rubik to teach his students to think in spacial terms. It consists of a cube that fits comfortably in the hand. This cube appears to be composed of smaller "cubelets" that stay together. Each face of the cube has 3 x 3 brightly colored cubelets. Each layer -- top, middle and bottom -- of the cube can be rotated 90 degrees in either direction to form another cube that is identical except that the colors are scrambled. Solving the puzzle consists of getting all the same colored cubelets on the same face of the cube.
The most famous mathematical puzzles are the unsolved puzzles. At a world conference in 1900, the most famous mathematician of the age, David Hilbert, proposed 23 puzzles that he would like to see solved in the coming century. These puzzles made careers and some of them spawned new branches of mathematics. All but one, the Riemann hypothesis, were solved. The Clay Mathematical Institute proposed seven puzzles--including the Riemann hypothesis--for this century, and it set up a $1 million prize for each puzzle. The Riemann hypothesis concerns the distribution of prime numbers--a result that will effect almost every branch of mathematics.