# Hat and River Problem

Below is the story problem:

A person dives from a bridge into a river and swims upstream through the water for 1 hour at constant speed. She then turns around and swims downstream through the water at the same rate of speed.

As the swimmer passes under the bridge, a bystander tells the swimmer that her hat fell into the river when she originally dove. The swimmer continues downstream at the same rate of speed, catching up with the hat at another bridge exactly 1 mile downstream from the first one. What is the speed of the current in miles per hour?

The answer given was 1/2 mile per hour and the author concluded it took 2 hours - "The whole thing took 2 hours, during which the hat traveled 1 mile downstream."

How did he come to the conclusion that it took 2 hours for the "whole thing"?

Give the provided information, we can't know the speed of the swimmer, but let's suppose the swimmer swims at a speed of $v$ miles per hour. That is, if the water would not flow, she swims at $v$ miles per hour. Since the hat flows with the water, that means that as the swimmer swims upstream, the distance between the hat and the swimmer increases at exactly that $v$ miles per hour. So, swimming back, the swimmer decreases the distance with exactly $v$ miles per hour as well. So, if she swam upstream for one hour, it'll take that one hour to catch back up.

The key to solving this puzzle is therefore to look at the movement of the swimmer relative to the river as a 'whole'. That is, if we look at the movement of the swimmer relative to all of space, the swimmer goes through space at an enormous speed, since the swimmer is located n planet Earth, which hurtles through space at a good clip ... and our solar system moves at great speed relative to the center of our galaxy, etc. etc. So thinking about these kinds of things, we recognize that 'speed' is always relative to something. Now, here on Earth, and for day to day purposes, most of the time we think of and measure speed relative to the 'surface' of the Earth. Indeed, that's is exactly the perspective you are asked to take when asked the speed of the river: that's relative to the river bank, or bridge, or river bed, or anything that's in a 'fixed' position as far as the earth surface goes.

OK, but for solving this puzzle, it is better to take think about the movement of the hat and swimmer relative to the river as a whole. That is, we can think of the river as a kind of 'snake' that moves, as a whole, relative to the Earth's surface. However, how does the hat and swimmer move relative to this 'snake'? Well, relative to the river, the hat remains at a fixed position; it's as if it is 'attached' to the snake. The swimmer, however, moves with a certain speed relative to this river; think of it as 'crawling' along the skin of the snake. And we can assume that is the same relative speed as if the swimmer is in standing water, i.e. in a lake or pool. And, when the swimmer turns around, the swimmer moves at exactly that same speed relative to the river again, just in the opposite direction. So, thinking about it that way, it really is analogous to the situation where she would dive in a lake, lose the hat right where she dove in, swim for an hour into the lake, and then turns around: how long does it take to get back to the hat? An hour of course!

• are you assuming that the speed remains the same for upstream and downstream? – ilovetolearn Apr 29 '18 at 13:13
• @youcanlearnanything I am assuming the swimmer swims with the same 'intensity', yes. Of course, from the point of view of someone standing on the riverbank, she goes faster downstream than upstream, but relative to 'the water' (imagine an individual water molecule flowing down, and how the swimmer 'swims away from it first, and then catches back) she swims with the same speed. Yes, that's what I assume. I also assume it takes $0$ seconds to turn around, and she is not slowed up talking to the person back at the bridge. – Bram28 Apr 29 '18 at 13:17
• @Bram27 can you explain how would you resolve it using algebra? – ilovetolearn May 1 '18 at 5:43
• @youcanlearnanything There are typically two phases to solving mathematical problems: First, put the problem into symbols. Second, figure out something by manipulating those symbols. For the second part we often use algebra, yes. But the first part is part and parcel of the mathematical solving process as well. In my Answer, I effectively did all the analysis to get it ready for symbolization. I'll do that symbolization now: We know that two hours past between the moment the swimmer lost her hat when diving in, and the moment the swimmer caught up with her hat. ... (continued) – Bram28 May 1 '18 at 13:05
• @youcanlearnanything During this time, the hat flowed down 1 mile. So, if we set the speed of the current to $v$ miles per hour, we have as an equation: $2v=1$. OK, we are now ready to do part two, which is using algebra to solve this equation: $v = \frac{1}{2}$. OK. Done with part two. The moral is: spending a bit of time doing part I, and thinking about how best to represent the problem can really pay off! – Bram28 May 1 '18 at 13:07
• $s$ - speed of swimmer, positive
• $r$ - speed of river, positive
• $p_1$ - position the swimmer goes upstream, positive
• $p_2$ - position the swimmer goes downstream, positive
• $t$ - the time it takes to go from $p_1$ to $p_2$

A person dives from a bridge into a river and swims upstream through the water for 1 hour at constant speed.

$$p_1 = (s - r)\cdot 1$$

As the swimmer passes under the bridge, a bystander tells the swimmer that her hat fell into the river when she originally dove.

$$p_2 = r\cdot (t + 1)$$

The swimmer continues downstream at the same rate of speed, catching up with the hat at another bridge

$$p_1 + p_2 = (s + r) \cdot t$$

exactly 1 mile downstream from the first one.

$$p_2 = 1$$

$$(s - r)\cdot 1 + r\cdot (t + 1) = (s + r)\cdot t$$ $$s - r + r\cdot t + r = s\cdot t + r \cdot t$$ $$s = s\cdot t$$ $$s = 0 \text{ or } t = 1$$

Since "catching up with the hat" wouldn't make sense if $s=0$, it must be that $t=1$.

$$p_2 = r\cdot(t+1)$$ $$1 = r\cdot(1+1)$$ $$\frac12 = r$$

Work in the frame of reference of the hat (i.e., the surface of the river). The swimmer swims away from the hat for one hour and then swims toward the hat until she gets back to it--which clearly also takes one hour.