be careful what you wish for

Today in calculus we were practicing substitution, a procedure by we which we are able to convert seemingly complex functions into simple ones in order to do integration.  If you haven't studied calculus for a long time, or ever, I just spewed out a bunch of gibberish, but no matter.  I have included a problem here for your perusal. Whether you understand it or not, this problem and others like it often result in the need for fractions. 

A student commented, "I hate fractions.  Why doesn't math only use whole numbers."  It was good that my back was to him as he said this, or he would have seen my fiendish smile. 

"But there is such a math," I replied.  "It's called discrete math.  Would you like to try a problem?" 

Of course he agreed and so I promised to post it online and offered a little prize to the first student to solve it.  You may recognise it. 

There once was a city through which flowed a river.  At the center of the river were islands.  The citizens had built 7 bridges so that they could cross these rivers and move from one part of the city to another.  The map shows the layout of the city, the river and the bridges.  The land is green, the river blue, and the bridges gold.  The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side. The walk need not start and end at the same spot.  Can you find this path? 

Be careful what you wish for...   

substitution photo credit
7 bridges photo credit