Wet Hats

I have had occasion in recent times to sit by a riverbank, sparsely populated by ducks and rowers which has led me, via a series of curious and unrelated leaps, to notions about making decisions - a course which I used to teach in a previous and less congenial lifetime. Any problem - in other words a series of events involving decisions - necessitates making a choice from the set of available alternatives at each stage. Therefore, our decisions will only be as good as the alternatives we have to select from. So much for the obvious. Being aware of the alternatives available to us has two immediate benefits : First, if we know all possibilities, we are in the best position to choose the most suitable for our purposes. Second, we may sometimes find ourselves in a situation where there is no rational way to decide between them so the best we can do is to make a blind or intuitive choice. Should our chosen course of action lead us to a dead end, we would at least be aware that we made a choice earlier in the design and another course is still available to us. A failure to investigate the existence of valid alternatives might be the difference between a messy or elegant outcome.  Sometimes, I tell my students to walk round a problem and a small door to the solution may present itself which might look very different to the more obvious main entrance.

This simple problem illustrates the idea that if we start from the most useful place, matters can be simplified greatly.

A man has a boat that moves at constant speed in still water. He starts on a boat trip, moving upstream at 4 km/h. After 15 minutes, he realises he dropped his hat the instant he started the trip, so he turns around to get it. When he finally catches up with his hat, he is 2 km downstream from where he began the trip. Assuming the turnaround time is negligible, how fast is the stream moving relative to an observer on the bank?

This problem was found in the question paper of a competitive exam where the examinee is required to solve it within forty-five seconds. Most people implicitly choose to attack the problem as if they were a watcher from the riverbank, hence getting tied up in algebraic knots. Instead, try forgetting about the speed of the boat; instead try sitting on the hat as the end point and working it from there.