One of the most complicated problems in math concerns the Koch snowflake, an infinitely expanding shape of equilateral triangles. Due to the way in which the shapes are drawn and laws of limits, the area of the snowflake can be defined, but the perimeter diverges to infinity. Another common example is the Monty Hall problem, which is based on the game show "Let's Make a Deal."
The Koch snowflake was introduced in 1904 by Helge von Koch. It is one of the earliest known examples of fractals. The snowflake is formed by starting off with one equilateral triangle. More equilateral triangles are placed on the outside of the original, but their sides must be equal to one-half that of the triangle they are resting on; after each iteration, the number of sides on the snowflake increases by a factor of four. Due to this, the snowflake is never completed and expands infinitely. The perimeter of the snowflake can never be solved and is technically infinity. However, since the triangles eventually become very small, the total area can be approximated to 160 percent the area of the original triangle.
The Monty Hall problem involves probability. A game show host asks the contestant to choose between three doors. One has a prize, while the other two contain some gag, traditionally goats. After the contestant chooses his door, the host reveals one of the goats. The contestant is then invited to either switch doors or stick with his original door. Due to probability, the contestant is mathematically more likely to pick the prize door by switching. In fact, he has a two-thirds chance of winning the prize.