Math Is Fun Forum

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Ants puzzle

Hi,

You may be familiar with this puzzle.

Consider a regular polygon with n sides, and side length 'a'.
There's an ant at each vertex. Every ant moves in one direction(all left or all right) towards the nearest ant in that direction, all at the same,constant speed. They move until they meet. What is the distance traveled by each ant?

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"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Ants puzzle

Hi gAr;

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Hi bobbym,

Thanks for the link.
I knew of the formula, and but cannot recall the site from where I read. That site too only cited the formula.
I tried hard to derive the formula, but couldn't. Then I posted it here!
I'm hoping someone may work it out here

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Ants puzzle

It is a pursuit curve. But I am having some trouble with it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Ok, thanks for letting me know the name.
I'll search for some materials related to it. It was difficult without knowing the keywords!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Hi,

There is a book dedicated to this topic.
"Chases and escapes: the mathematics of pursuit and evasion"
By Paul J. Nahin
You may check few pages in google books

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Ants puzzle

Hi gAr;

I believe I have the book in my closet under a ton of others. I only went through a little of it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Wow! You seem to be having a good collection of books.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Ants puzzle

Best in the city! Now getting through them is another matter.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Do you have a library at home or do you live in a library?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Ants puzzle

Sometimes it seems like both.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Ants puzzle

Hi gAr,

Consider three ants at the end of an equilateral triangle of side length = s moving towards each other at constant speed = v.

As the ants move towards each other, points joining the three ants will be an equilateral triangle of a smaller side.(by symmetry)
As time passes the side of this equilateral triangle decreases.

If you choose any of the ant as your frame of reference, the component of velocity of the other ants towards the fixed (frame) ant
will be v + v/2. where v is the speed of the ant. [because velocity of A wrt. B is equal to  velocity of A wrt. F +  velocity of F wrt. B].
This means that the fixed ant will see the others coming closer to it at speed = 3/2v.

And in the ant frame the other ants travel towards the fixed one in a straight line. Therefore distance traveled(in ant frame) = s.

Time for which the ants move is = T = dist/speed = s/(3v/2).

In the ground frame they move with speed v for a time = T. Therefore distance traveled by the ants = vT = (2/3)s

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Hi 123ronnie321,

Thanks for the explanation.
Did you check: http://mathworld.wolfram.com/MiceProblem.html

Actually, I wanted to know whether there's any easy derivation for a regular polygon with n sides.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

123ronnie321

Member

Registered: 2010-09-28

Posts: 128

Re: Ants puzzle

That same logic can be applied to an n sided polygon. I have not seen any shorter method for this puzzle.
The above solution is similar to that given in the book 'Concepts of Physics' by H.C. Verma. Have you seen that book?

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Ants puzzle

Hi,

Okay.
No, I haven't seen that book. I'll check.
Thank you.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."