The Seven Bridges of KonigsbergIn Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this:
The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.Problem 1
Try it. Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke. Solution

Problem 2
Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:

Now try to solve the problem. SolutionProblem 3
Does it matter which bridge you take away? What if you add bridges? Come up with some maps on your own, and try to 'plan your journey' for each one.

Historians estimate that by 2000 B.C. humans had noticed that the ratio of circumference to diameter was the same for all circles. This discovery hinged on the idea of proportion - in this case humans noticed that if you double the distance "across" a circle, then you double the distance "around" it. In today's algebraic notation this implied the formula

where Pi was constant. (It wasn't until 1706 that this notation, using the Greek letter seen in the above equation - often written Pi and pronounced like the English 'pie' - was introduced by William Jones).The significance of this discovery is clear: Circles are everywhere - in the sun, the moon, the pupils of our eyes, the most basic religious rituals and the earliest man-made structures. Achieving a greater mathematical understanding of Pi would lead to scientific and technological advances that would further the development of civilization, as well as creating some very interesting problems in pure mathematics.
But one problem remained - what is the numerical value of Pi?

The Seven Bridges of KonigsbergIn Konigsberg, Germany, a river ran through the city such that in its center was an island, and after passing the island, the river broke into two parts. Seven bridges were built so that the people of the city could get from one part to another. A crude map of the center of Konigsberg might look like this:The people wondered whether or not one could walk around the city in a way that would involve crossing each bridge exactly once.

Problem 1Try it. Sketch the above map of the city on a sheet of paper and try to 'plan your journey' with a pencil in such a way that you trace over each bridge once and only once and you complete the 'plan' with one continuous pencil stroke. Solution

Problem 2Suppose they had decided to build one fewer bridge in Konigsberg, so that the map looked like this:

Now try to solve the problem. Solution

Problem 3Does it matter which bridge you take away? What if you add bridges? Come up with some maps on your own, and try to 'plan your journey' for each one.

But one problem remained - what is the numerical value of Pi?