Casey's Cow and the PONR

I go for a four mile walk every day to keep everything working and it's during those walks that my brain goes off in all sort of weird directions. Today there was one point where I was walking down a narrow country lane and a car was coming towards me. I could see a gap in the brambles up ahead, so broke into a run so I could duck into the gap and let the car past. And it got me thinking of Sam Loyd's Casey's Cow problem.

The problem is all about a cow on a railway bridge. The bridge is d meters long and the cow has walked x metres along it. A train appears in the distance y metres beyond the end of the bridge. The cow can run at speed u and the train travels at speed v. Should the cow run towards the train or away from it if it wants to live? Sam's original problem had letters rather than numbers and you had to solve for one of the letters but I prefer this version of the problem with letters everywhere and the problem being to derive a decision rule.

At first this looked scary with five letters flying around suggesting I'd need to be plotting all sorts of hyperplanes in five dimensional space. But I quickly realised this was an illusion. It's only a three dimensional problem, with those other two dimensions being the choice of units. I'm going to make up a new unit of length. Let's call it a deltoid and I'll define a deltoid as being the length of the bridge. So d disappears and if I measure other distances in deltoids, x can be thought of as the proportion of the bridge that's already behind the cow. Then I need a new time unit, the snickett. A snickett is how long it takes a cow to run one deltoid. So u=1 and v is effectively train speed divided by cow speed.

So, anyway, on to individual strategies. For running towards the train to work, the cow needs to cover 1–x before the train travels y, so 1–x<y/v or y>v–vx This divides up the xy plane into two areas where this tactic either works or doesn't:

And for running away from the train to work the cow needs to cover x before the train travels y+1, so x<(y+1)/v, so y>vx–1 and we divide up the xy plane into a different two areas:

Put those two graphs on top of each other and we get something that divides the plane into four areas: one where both tactics work, one where both fail and two where one works and the other fails:

What makes this interesting is that point where the two lines cross. At this point, x is (v+1)/2v and y is…well I don't really care. Because (x+1)/2v is the Point Of No Return, PONR. If x>PONR, then there are no scenarios in which running away from the train works and running towards it doesn't. And vice versa if x<PONR. So a simple rule of thumb would be to run towards the train if the cow has gone past the PONR and to run away from it otherwise. There will be scenarios where the the cow does either way or lives either way but the cow can never get a better outcome from following a different strategy.

So what started as a five dimensional problem ended up reducing down to a really simple rule, with a PONR expressed as a proportion of the length of the bridge and depending only on the ratio of the cow's and train's speeds and not on the distance of the train from the bridge. I was amazed that it could all be shrunk down so neatly.

As for practical applications, if a car travels narrow lanes at, what, 5 times the speed I could run at, then plugging in v= 5 gets me to a PONR of 0.6. So from now on when I'm out walking on narrow roads and walk past a gap in the brambles, I'll be looking ahead to the next gap, mentally marking out a PONR 60% of the way towards it and knowing that's the point beyond which I'm better off running towards the car. And it doesn't matter how much warning I get about a car coming the other way.

Anyway, that kept me occupied for the first half of my walk. For the second half, my mind was occupied with more important stuff.