Do you ever listen to the radio program Car Talk? Often they share some interesting mathematical problems.
But, one day (6/18/2011), a problem arose that they did not even recognize. A caller was trying to resolve a quandary.
She lived next to a hill, and claimed that two roads led to the top of the hill. One road was straight up the hill...at what she initially claimed was at a 60o incline. The second road wandered back and forth in a horizontal manner involving minor slopes as it gradually meandered to the top of the hill.
Now my concern. The caller claimed she had driven both roads and found them exactly the same length of 3.1 miles....and wanted to know which route used up more gas.
You may ignore the question (and the steep incline)...but I hope you caught the geometrical impossiblity being described. The two roads cannot be the same length, even if she lived in a non-Euclidean world. It is all covered by the definition of a straight line as the "shortest distance between two points."
Yet, the show hosts did not say anything about this difficulty...which suggests they only recognize something as a mathematical problem IF you preface it with: "Consider this problem..."!