[Eternal Student (OP)]

Hi.

   Are you bored?  Do you want a short puzzle?  No?  Well here's one anyway.   
It doesn't require much mathematics, I reckon someone who's studied some maths or physics at post 16 years of age should have all they need to solve it.
- - - - - - - - - - - -

Four dogs are in four corners of a square of side length 1. Each dog starts running towards the dog immediately anti-clockwise to it. The dogs start at the same time, they all run at the same speed, and at every moment each dog is running directly towards the neighbouring dog.

During the pursuit, the dogs will run in a spiral before they all meet in the centre. How far does each dog travel before the group collision?

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Puzzle taken from an article published in the Guardian:   https://www.theguardian.com/science/2019/jun/03/can-you-solve-it-dogs-in-pursuit
   BUT DON'T FOLLOW that link unless you want to see the solution.   

Best Wishes.

[chiralSPO]

Are these dogs point particles that must be co-localized at the endpoint, or do they have some nonzero radius that is less than the "1" specified as the side length?

[Origin]

Are these dogs point particles

LOL, good question...

[Halc]

The calculus appears pretty complicated, but the picture shows some trivial facts.
First, to answer a prior question, I assume everything is a point, the problem otherwise not really being fully defined.

You've approximately shown nearly half the distance traveled. At the actual point (har!) the distances between them have been halved, the answer is simply twice the distance already traveled.

The remaining distance is a square still, but rotated something like 30 degrees or so. Each time the distance is halved, the square gets rotated by that amount, so the dogs are going to circle each other an infinite number of times before the collision after a total path length.

My attempt: The problem can be broke into tiny pieces, so let's say a dog moves a trivial distance x, almost a straight line. It will curve just a little bit, but for sufficiently small x, a straight line, bringing it exactly x closer to the next dog. This leaves the same initial condition, but rotated and a little closer. Therefore the path length of each dog is exactly 1 by symmetry since each movement of x reduces the distance between them by x.

[Bored chemist]

Spoiler alert

Spoiler: show

Halc doesn't seem to know how to use spoiler tags

:-)

[Petrochemicals]

Is the answer they will never reach.?

[Eternal Student (OP)]

Hi.

1.   Yes they are point particles.
2.   Halc's approach is perfectly reasonable.   Calculus could be used but it really isn't required in this instance.   Halc's second approach is just a minor variation on the solution recommended in the original article, so well done Halc.   I suspect the explanation given on the website is just slightly nicer - but only for people who already know something about velocity and vectors (hence I estimated study post-16 years of age).
3.   There's a full discussion on the Guardian website that was linked to in the first post,   the first page is just a discussion but there is a link from there to another page with a full solution.

   I've never tried using a spoiler.... so let's see if I can copy and paste the most relevant bit in here:

Spoiler: show

   The image of the four spirals, below, helps understand why. At each moment on their paths, the dogs are at four points of a square. The direction in which a dog is headed is therefore at all times perpendicular to the direction in which the dog that it is chasing (and which is chasing it) is headed.

    Consider the view from one of the dogs (the chaser). It is chasing a dog (the chasee) that at every point is moving perpendicular to the line between them. In other words, there is no part of the velocity of the chasee that is taking it further away, or closer to, the chaser beyond the movement of the chaser. As far as the chasing dog is concerned, the chasee may as well be standing still. The length of the total path travelled therefore is the distance between chaser and chasee at the start, which is the side of the square.

Best Wishes.

[Eternal Student (OP)]

Hi again.

Is the answer they will never reach.?

    Interesting.  The situation is idealised, in reality the dogs probably can't turn arbitrarily sharply, so they might end up chasing each other around in a circle.   However, keeping the problem simple and idealised, they do actually meet in a finite amount of time.
    This also relates to something Halc said:

....so the dogs are going to circle each other an infinite number of times before the collision after a total path length....

    Yes.   It's another example where an infinite set of things happen within a finite amount of time (and distance travelled).

   I'll see if I can find another short puzzle with dogs in a day or two.

Best Wishes.

[Petrochemicals]

Hi again.

Is the answer they will never reach.?

    Interesting.  The situation is idealised, in reality the dogs probably can't turn arbitrarily sharply, so they might end up chasing each other around in a circle.   However, keeping the problem simple and idealised, they do actually meet in a finite amount of time.
   

Given that each dog is on the inside of the dog it chases, logic would dictate that the dogs would catch each other before they reach the centre, that somehow one dog will overtake the other even though they are travelling at the same speed. The theoretical idea of infinite curved paths seems like it relies on other factors, if it was limited only by velocity, when the rate of change equals the dog speed I would think they would be chancing their tails. This is a science forum which just loves things like centrifugal force and ceptripedal accelleration, gravity, free fall etc.

[Eternal Student (OP)]

Hi.

Given that each dog is on the inside of the dog it chases, logic would dictate that the dogs would catch each other before they reach the centre, that somehow one dog will overtake the other even though they are travelling at the same speed.

    The ability of the chaser to turn "inside" of the dog they were chasing is what made it possible for them to collide, if that helps.   If all the dogs were forced to follow one track (perhaps the perimeter of the original square) and couldn't just cut across and aim directly for the other dog then they would never have caught each other because they do all have the same speed exactly as you stated.
    The dogs had no idea they would end up at the centre.  They were always just aiming directly for the other dog.   This makes it a bit different to a typical human athletics race where the runners were actually aiming for a fixed position (the finish line) and having an inside track will help because the judges will blow a whistle (or shoot the offender with the starting pistol) if another runner just collides into you.

Best Wishes.

[Eternal Student (OP)]

Hi.

Here's a new short puzzle with dogs:

      You are standing at the centre of a circular field of radius R. The field has a low wire fence around it. Attached to the wire fence (and restricted to running around the perimeter) is a large, sharp-fanged, hungry dog. You can run at speed v, while the dog can run four times as fast. What is your running strategy to escape the field?

   That's it, that's the original wording of the puzzle and it's shorter and neater left this way.   Apply reasonable common sense to answer all the minor questions about how and why the dog moves.   I'll put some details under this spoiler if you want clarification - warning these details might lead you in the direction of a solution.   It would be better if you tried to imagine what is happening yourself first and only then check the first spoiler.

Spoiler: show

(i)   The dog is attached to the wire fence by something like a short piece of rope.  Ignore the length of the rope, it's short.  The person who tied the rope was careless and allowed it to slide along the wire, that's why the dog can run around the whole perimeter.
(ii)  The general idea is that the dog is keen to eat you.  This will happen if you and the dog end up in the same place (so that will be somewhere on the perimeter of the field).
(iii)  The dog will run around the perimeter so as to keep the distance between you and the dog as small as possible.  The dog doesn't have to keep running all the time, for example it will stay stationary if it is already as close to you as it can get.
(iv)  The dog can turn to run the other way around without any significant delay if required.

This puzzle was found on this website:    http://puzzles.nigelcoldwell.co.uk/thirteen.htm
   BUT DON'T FOLLOW  that link unless you want to see the solution.

There's a significant hint for a solution under this spoiler:

Spoiler: show

  Do you feel like you're often running around in circles?   

Best Wishes.

[Halc]

To get this straight, the nasty dog is restricted to the circular path and always moves at 4x your speed in the direction that takes it closer to the point on the circle to which you are nearest.

Seems pretty simple.

Spoiler: show

Run away from the dog. As it picks a direction, you change course and follow a circular path as well. This works fine as long as you're within a quarter of the distance from the center to the edge. So you are capable of keeping the dog at max distance until this point.  Starting at that point, you break for the edge, and angle your path in a 1-4 ratio from the dog's motion, just to buy a little more space.

Is this enough?  Circle of radius 1 and the dog needs to travel 3.14.  I am at 0.25 and need to travel 0.75.  No problem. I don't even need to do the angle bit.

I think a curved path might be optimal and it would seem an interesting problem to determine the max speed that the dog could run before you could not escape the circle. I suspect it's over 5x, but didn't work it out.

[Annieclo]

HAHAHAHA:) This is really interesting game..

[Eternal Student (OP)]

Thanks @Annieclo
   I'm glad you enjoyed the puzzle.

[Petrochemicals]

I believe this is the reverse of the previous conundrum, 1/4 perimeter speed, circular rotation, etc.

[Eternal Student (OP)]

Hi.

I believe this is the reverse of the previous conundrum, 1/4 perimeter speed, circular rotation, etc.

   Wow,  I wish I had been that smart.  I can see why the problems do look like they might be related or "dual problems" but actually I haven't done that deliberately.
   It took me long enough to find another puzzle that involved dogs and that's the only link with the previous puzzle I was going for.

Best Wishes.

[Bored chemist]

I believe this is the reverse of the previous conundrum, 1/4 perimeter speed, circular rotation, etc.

What problem is the reverse of what problem?

[Eternal Student (OP)]

Hi.

What problem is the reverse of what problem?

   None of them.

I can see why the problems do look like they might be related or "dual problems" but actually I haven't done that deliberately.

Best Wishes.

[Eternal Student (OP)]

Hi.

   An earlier post from someone has been seriously edited to become what appears to be an advert.  Although the original message was marked with a "thanks" or similar annotation from me, it was not my intention to promote a casino.  That "thanks" annotation has been withdrawn.

Best Wishes.

[Petrochemicals]

Hi.

What problem is the reverse of what problem?

   None of them.

I can see why the problems do look like they might be related or "dual problems" but actually I haven't done that deliberately.

Best Wishes.

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