[Eternal Student (OP)]

Hi.

It probably is time to make the solution available.
   Fortunately that's easy.  You can just go and read Halc's earlier post and uncover the spoiler they put there.

   There are many possible solutions or ways to escape from the dog.  There's a bit more of a discussion here on the website where the puzzle came from:
http://puzzles.nigelcoldwell.co.uk/thirteen.htm
I'm very cautious about suggesting anyone should follow a link - You DO NOT have to.  Halc's explanation is good enough and you can just read that off this website.   The link is here mainly because it's only fair to acknowledge the original author etc.

Approach the dog keeping just out of reach and beat it to death with a stick.

     That's a possible solution but not one I would recommend.  Anyway, I never said there were any sticks in the field.

I don't think I know any more short puzzles with dogs but I'll see what I can do in the near future.

Best Wishes.

[Petrochemicals]

Hi.
.

Approach the dog keeping just out of reach and beat it to death with a stick.

     That's a possible solution but not one I would recommend.  Anyway, I never said there were any sticks in the field.

You never said that there were not.

[Eternal Student (OP)]

Hi.

   It might be time for a new and suitably short puzzle with dogs.

   Find the shortest route for a thirsty dog going home.

The dog starts  100 m  west of his home and  10 m North of it.    There's a river running west to east which is 30 m south of his home.   The dog must get home and get to river at least once on the way.    What is the shortest route?


Just to help I've given you a diagram.... but you should really draw it yourself.   Here's a possible route, not necessarily the shortest...

dog-river2.png (44.03 kB . 1121x549 - viewed 927 times)

Your task is to describe the shortest route.

Best Wishes.

(LATE EDITING:  Extra credit if you don't use calculus.)

[Halc]

I'm still working on the circle problem, but have had almost no time to do so. It's coming.

Maybe the dog can shorten the effort by jumping on a board floating on the river, saving steps. I presume such complications are not part of the problem.
This problem has a physical solution:

Spoiler: show

Just put pegs in a board where the house and dog are, and a rod representing the river. Tie a string to the dog and the house looped around the rod and pull it tight. It will move to the shortest path, which momentarily touches the river 4/7th of the way, or ~43 m west of the house.

[Halc]

Another physical solution:

Spoiler: show

Replace the river 'rod' with a mirror and shine a laser (a surveyors laser that leaves a line on the ground) from the dog to the reflection of the house

[Eternal Student (OP)]

Hi.

   Well that is a fast response, @Halc .
I like the practical approach you suggested with the string.  It obviously works this time but technically it only shows that you have the shortest route available which doesn't require any intermediate lengthening first.   Or to say it another way, you have reached a local minimum not necessarily a global minimum.

   I'm not going to comment on the laser idea, it's ok but we'll leave it under the spoiler for a day or two.

Best Wishes.

[Halc]

you have reached a local minimum not necessarily a global minimum

OK, you wait until a flood occurs and the water reaches just to the straight line between dog and house.
If that's on the table, you might as well wait for the flood to get even worse and perhaps carry the house much closer to the dog, which isn't too far off from my 'riding the raft' idea.

[Colin2B]

Ok, I’m limited on wifi time available so complete guess (based on other physics)
I describe route by 2 straight lines, one towards the river which when hits river 2nd line goes to home. Angles on either side of perpendicular to river have to be the same.

[Rolerwill]

In theory the will never reach each other, IRL it depends when one dog starts increasing its speed or the other(s) start decreasing it.

[Eternal Student (OP)]

Hi @Rolerwill ,

    I'm not sure I've spoken to you before.  Welcome to the forum.
I think you might be talking about the first ever puzzle discussed on this thread.    There are some answers already presented for that.   It's also just an idealised problem and in real life things would get in the way.   If you're interested, the discussion and solutions presented by the Guardian newspaper will probably save time rather than reading through the whole of this thread. 
   https://www.theguardian.com/science/2019/jun/03/can-you-solve-it-dogs-in-pursuit  .

Best Wishes.

[alancalverd]

Each dog starts running towards the dog immediately anti-clockwise to it.

Apologies for not addressing the problem mathematically, but the experimental scientist knows that whilst small children follow a pursuit curve, dogs are actually better hunters and tend to run in a straight line towards a predicted intercept. Eventually, some children learn the trick and turn into good cricketers or footballers. David Beckham's exceptional ability as a midfielder was being able to make long passes to an intercept so that wingers and strikers could run on to the ball at full speed in a straight line.

Late edit: it's also the art of a fighter controller!

[Eternal Student (OP)]

Hi.

   It's probably time to discuss the solution.  I first saw this puzzle (not sure it had a dog) in a magazine produced by Imperial College, London University where the writing team had spent a little time examining the approaches used by various undergraduates of different disciplines.

One Mathematician started at #0,   most Mathematicians and Physicists started at #1,   most others tended to skip straight to #2.   Less than half the people of any discipline used #3  (they used calculus instead).

0.    Banish the infinite  A finite length route exists (e.g. one is shown in the digram with the question).  All routes we consider from now on behave sensibly.  All routes we consider have finite length between all points along the route (else it can't be the shortest route).

1.    Establish that the dog only goes to the river once   
    If it intersects the river at two (or more) points, R₁ ≠ R₂  then pick any one point, R₁.   Just draw straight lines from the start to R₁ and from there to the dog house.   That new route satisfies our criteria and cannot be longer than the old route - just by the basic definition of a straight line in Euclidean Geometry as the unique shortest path between two points.   It's possible to show that the new route is actually shorter but only a few Mathematicians bothered, most people just claimed it.  The basic proof is obvious... the dog must get back to river at R₂ so they deviated from the straight line sections somewhere.  (We did actually need step #0, if the point R on the river was infinitely far away then the dog can run along the river for a while and that's perfectly co-linear with straight line sections to the dogs home etc.)

2.   It should be clear that every finite route can be reduced to two straight line sections with three points of interest, the start (S), the point (R) on the river and the dogs home (H) at the end.   These are the only contenders for a shortest route we ever need to consider again.   Points S and H are fixed, the only issue is where you put the point R along the river.
    At this stage, over half the people (regardless of their discipline of study) continued by using calculus.  Set up a total distance function D with a single variable, x, which is the x co-ordinate of the point R along the river.   Find the local minimum.   That works.   It only finds a local minimum but you can improve on this and establish it's a global minimum just by sketching the function  etc.   (I don't think many people actually did bother).

3.   It's not necessarily "better" and it might actually take just as many lines to explain but you can, of course, solve this without calculus.

SRH.png (8.18 kB . 1130x792 - viewed 771 times)
    Just consider a related problem which is finding the shortest path from S to  H'  that crosses the river at least once.  Where H' is another dogs home located so that it is the reflection of the proper home H in a mirror along the river.  Fill in the details... Just as in the original problem it reduces to placing a point R on the river and straight line sections from S to R and from R to H'.    The total Distance D for the original problem =  SR + RH,  while the total distance for the reflected problem is SR + RH'    but  due to the triangles being identical   RH = RH'   and so these total distances are always the same.  Considering optimal solutions for either problem amounts to nothing more than varying the position of R along the river and evaluating the associated total distance D (or the identical value of D').
     Proving the existence and uniqueness of an optimum solution for the reflected problem is easy...  any path from S to H' will cross the river at least once.... so that problem reduces to find the shortest path from S to H' .   By Euclidean geometry, that's a straight line.  ∴ R can only be placed at one point on the river for an optimal solution to the reflected problem.
    That's it,  dot the i and cross the t .   The existence and uniqueness of an optimal solution for the original problem follows.
   Simple geometry will then tell you where R must be placed,   @Halc  said ~43m west of the house,  @Colin2B  said to use the angle of incidence = angle of reflection.   Well done to those two and especially for spotting that reflection would help instead of going straight for the calculus.

- - - - - - - - - - -
    If you were developing these ideas, you could have the dog go for a swim in the river and be slower when wet.  To solve these problems you need a woofraction index, a ratio of speeds while wet and dry.

Best Wishes.

[alancalverd]

I've always wondered whether "proper" mathematicians allowed the use of obvious symmetry and reflection as tools in a formal proof. I was so inspired in my youth by Kasner and Newman's "Mathematics and the Imagination" that I married the only girl I ever met who shared that enthusiasm (both, it turned out, recommended by our electrical engineer fathers). Like a good novel, it began with a surprise - the use of obvious symmetry to simplify a classic proof.

The dog and river problem actually has a practical application in navigation. If you are sailing or flying to a visible destination with an unknown cross wind or tide, simply aiming directly at the target will take you on a spiral pursuit curve, arriving pointing directly into wind. The trick is to maintain your original compass heading for half the distance, then turn through four times the difference between that and the new bearing. It's oddly counterintuitive but definitely optimal.

[Petrochemicals]

Hi.

   It might be time for a new and suitably short puzzle with dogs.

   Find the shortest route for a thirsty dog going home.

The dog starts  100 m  west of his home and  10 m North of it.    There's a river running west to east which is 30 m south of his home.   The dog must get home and get to river at least once on the way.    What is the shortest route?

Straight to the dog house and vertically down to the river.

[Eternal Student (OP)]

Hi.

I've always wondered whether "proper" mathematicians allowed the use of obvious symmetry and reflection as tools in a formal proof.

     At school, sure no problem.   You just had other problems to worry about, like whether an unusual proof would actually be on the mark scheme.   I mean, if there's a small error then even if the marker did spend time to examine what you tried, if it wasn't on the mark scheme and what can they do?

     At university, it depends on whether you were doing Pure Mathematics or Applied mathematics.   It general, it's OK and potentially very good - but there might be a shift in how you present the information.  In pure maths there are numerous things that are fairly obvious results when described in plain English but take undergraduates a week's coursework to set up and prove formally.   If you want to keep a Pure Mathematician busy, ask them if "this result" can be proven without Euclidean Geometry.

....that I married the only girl I ever met...

    That bit was well written.  Congratulations, it sounds amazing.

Straight to the dog house and vertically down to the river.

   I guess that might work.   The original problem did ask you to find the shortest route for a thirsty dog to get home.  The dog did get home and it was thirsty.

Best Wishes.

[Petrochemicals]

Straight to the dog house and vertically down to the river.

   I guess that might work.   The original problem did ask you to find the shortest route for a thirsty dog to get home.  The dog did get home and it was thirsty.

Best Wishes.

It did, and it is possibly the shortest distance to answer the riddle. Conversely with the dog being thirsty it may be the quickest (shortest duration) for the dog to arrive immediately at the river and then toddle off home as the dog probably will function far better once refreshed.

[Bored chemist]

What sort of "home" for a dog doesn't provide water?