# Problem of the Day: 1/20/13

If seven evenly matched competitors run a race, in how many ways can they place 1st, 2nd, and 3rd?

Solution to yesterday’s problem:

Let’s use an annotated version of the diagram:

Knowing that the side length of the total square is 2 ft, we can say that the segments to either side of the octagon’s sides are 0.5 ft each. This makes four triangles, each with a base and height of 0.5 feet. Expressing their combined area gives

$A=4[\frac{1}{2}(0.5)(0.5)]$.

Simplifying,

$A=2(0.5)(0.5)$

The final answer is that there is 0.5 square feet left in the square.