We make a long list of numbers in the following way. We start with \(1\) and \(1\). After that, each new number is the last digit of the sum of the two numbers right before it. For example, the beginning of the list is \[1,\,1,\,2,\,3,\,5,\,8,\,3,\,1,\,4,\ldots\]
Show that, if we keep making numbers like this forever, the list must eventually start repeating in a loop.
In Problemtown there are \(n\) farms and also \(n\) wells which we think of as points on a plane. We know that no three points lie on a straight line. The mayor wants to build straight roads so that each farm is connected to exactly one well, and each well is connected to exactly one farm. The mayor insists that no two roads are allowed to cross each other. Prove that this is always possible.