Math Is Fun Forum

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Bitter areas.

Hi;

3 listening posts are situated in the north atlantic. They form an equilateral triangle. A nuclear explosion takes place 500 miles from one listening post, 1200 miles from another and 1300 miles from the last listening post. The explosion takes place inside the triangle that the posts form. What is the area of the triangle that the listening posts form?

A says) Impossible to determine, Cauchy wrote a paper on it and I am well familiar with that paper. It is a great advantage that I enjoy knowing more theory than B does. I know when a problem can not be solved, B does not. That is why I can work on the ones that can be solved.

B says) Hold on A, the problem is easy using...

C says) Cauchy hmmm, I have heard of him.

D says) Yes, he owned a bakery on my block and made good cannoli.

C says) Weren't they first discovered by Schiaperelli?

A says) Yes, I read a paper about the cannoli's on Mars, they were found to be a hoax. Something B probably knows all about and believes.

E says) What the heck are you talking about? I hate idiots and A,C and D qualify. I have already solved the problem.

Can you?

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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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi Bobby,

Is it...

Last edited by phrontister (2015-02-07 00:58:08)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

---

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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi;

Last edited by phrontister (2025-12-11 14:09:58)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

---

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.

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi;

Last edited by phrontister (2015-02-08 03:10:34)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

---

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.

Matir

Member

Registered: 2016-10-08

Posts: 22

Re: Bitter areas.

Hey, isn't the Earth round?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Not really, the spin distorts the shape. For the sake of the problem a flat plane was considered okay.

---

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Bitter areas.

Well. I am getting analytically the same result as phrontister. There is also a solution for the blast to occur outside the triangle formed by the posts.Then the area will be

sq.miles.

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

If you got the same answer as phrontister you have solved the problem. Good work. The other answer violates the terms of the problem.

---

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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

Hello:

Quite elementary.

Last edited by ElainaVW (2016-10-09 21:10:39)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Bitter areas.

Hi;

Yes, that is a solution and it makes good use of the new Region commands. Very good.

---

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.

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

How do you say it, yall gots to come up with something a bit harder aye bumpkin.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Bitter areas.

Hello:

Quite elementary.

Is it elementary or elementri containing 3 Elements?

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

ElainaVW

Member

Registered: 2013-04-29

Posts: 580

Re: Bitter areas.

Hello:

How about too easy. That will take care of tary-try-tree-tri-trey.

mycalsuite

Member

Registered: 2025-09-10

Posts: 81

Website

Re: Bitter areas.

A is right; there isn't enough data to identify the area in a unique way.

The equilateral triangle's size is not uniquely fixed by the distances to its three vertices.
As a result, the area cannot be calculated using the information provided.

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mycalsuite

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi mycalsuite;

A is right; there isn't enough data to identify the area in a unique way.

The equilateral triangle's size is not uniquely fixed by the distances to its three vertices.
As a result, the area cannot be calculated using the information provided.

By "not uniquely fixed", do you mean that multiple solutions are possible?

If so, do you have an example you could post?

Bobbym (OP), thickhead, ElainaVW and I have all arrived at the same single solution (ElainaVW's 2500√3(169+60√3) being slightly less simplified than the others' 2500(180+169√3).

Here's the Geogebra drawing I mentioned in post #4:

The Geogebra measurements check out...

EDIT: I googled this puzzle yesterday, and found that it has appeared in different guises over many years (more recently on websites such as Artofproblemsolving, math.Stackexchange and Reddit). The three internal-point-to-vertex lengths are often a Pythagorean triple (bobbym's {500, 1200, 1300} is one), opening up the use of a particular solution strategy. All solutions I found are single.

Last edited by phrontister (2025-12-11 13:20:12)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi mycalsuite;

The three internal-point-to-vertex lengths are often a Pythagorean triple (bobbym's {500,1200,1300} is one), opening up the use of a particular solution strategy.

Here's a geometric solution...

1. Draw large △ABC, consisting of 3 smaller coloured triangles △BAD, △CBD & △ACD that connect to inner point D where the explosion occurred.

2. (a) Rotate the 3 coloured triangles 60° clockwise:
          (i) the gold around point B, producing point E and forming the 60° ∠EBD.
         (ii) the green around point C, producing point G and forming the 60° ∠GCD.
        (iii) the blue around point A, producing point F and forming the 60° ∠FAD.
    (b) These rotations together form outer hexagon AFBECG, whose total area is exactly twice that of △ABC.

3. Draw lines DE, DG & DF to complete triangles △DBE, △DCG & △DAF, respectively.
   (a) These 3 triangles are equilateral as they're isosceles with a 60° included angle at B, C & A, respectively.

   (b) The remaining regions of the hexagon — the 3 triangles △DEC, △BDF & △GAD — are congruent right-angled triangles, each with sides {500,1200,1300} — a Pythagorean triple.

4. The area of the hexagon is the sum of:
   (a) the areas of its 3 equilateral triangles, and
   (b) the areas of its 3 congruent right-angled triangles.

5. The area of △ABC = ½hexagon = 2500(180+169√3) ≈ 1181791.46619785.

Last edited by phrontister (2025-12-11 13:12:32)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,808

Re: Bitter areas.

hi Phro,

I'm getting a 'Content Not Viewable In Your Region' message instead of your Imgur images. Any idea why that is happening?

I've checked back on all your posts and old posts back to June '24 are also producing this message.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi Bob,

I'm getting a 'Content Not Viewable In Your Region' message instead of your Imgur images. Any idea why that is happening?

Yes...but only after googling it a few moments ago.

From Reddit.com: Imgur blocks access to UK users after proposed regulatory fine.

Not sure exactly what to do about that from down here, but Reddit gives some suggestions for UK-ians: Imgur alternatives for UK users.
Well...that's all VERRRRRRY annoying!!!!!!

EDIT: I'm trying another file host: https://postimages.org/, and have changed the images in this thread over to them. Do my images show up for you now? There are 5 in post #19, 1 in post #18, & 1 hidden one in post #4.

Last edited by phrontister (2025-12-13 19:48:08)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi Bob;

I started using Imgur in late Feb 2017, and have 304 images stored there.

Most of them are on MIF posts, and, from what you've found and from what's reported on Reddit and elsewhere about Imgur's UK ban, they're probably all now invisible to UK members.

I did some testing down here via a VPN set to several different UK locations, but that failed as all my images (over a large range of dates) remained visible to me.

Maybe I'll cherry-pick a post or three to 'fix' my invisible images for UK-ians by changing them over to a non-Imgur file host...like I've done in this thread.

I wonder if Imgur's UK ban is permanent?

Of course, while the ban is in place, I won't be using Imgur for a global audience.

Last edited by phrontister (2025-12-13 11:16:58)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,808

Re: Bitter areas.

hi Phro, Thanks for the info. I was totallly unaware of this. I've followed the news about the UK Government's Online Safety legislation; I hadn't picked up on this side effect.

I've checked all my posts and I'm blocked from viewing every Imgur image!  There's loads of them. I have copies on my computer but I haven't got the time or enthusiasm to switch them all.  If asked by a poster I'll attempt to find my copy and get it on the site.

I haven't had a chance to check yet, but as I read it, I won't be able to log in to Imgur either.

What site are you now using?

And how long before that bans the UK too.

I'll give VPN a try when I get a moment.

Bob

ps. This could destroy us once the regulator gets to hear about us.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,918

Re: Bitter areas.

Hi Bob;

I've checked all my posts and I'm blocked from viewing every Imgur image!..........I haven't had a chance to check yet, but as I read it, I won't be able to log in to Imgur either.

From Reddit.com's Imgur blocks access to UK users after proposed regulatory fine:
     A help article on Imgur's US website, seen by the BBC, states that "from September 30, 2025, access to Imgur from the United Kingdom is no longer available".
     "UK users will not be able to log in, view content, or upload images. Imgur content embedded on third-party sites will not display for UK users."

What site are you now using?

https://postimages.org/, and so far, so good (well, nearly). Some points:
1. Registering enables using galleries.
2. Their hotlinks don't include a BBCode img-tag option. Best options:
    (a) 'Direct link' is missing the img tags...so enter them manually.
    (b) 'Hotlink for forums' includes the img tags, but also url tags...so delete them manually.
3. Image resizing is possible (but limited), and must be done before upload (Imgur's is after upload, making results easier/faster to analyse). Also, their resize compression spoils the image somewhat, and so I pre-sized the post #19 images with IrfanView (my image viewer) for better quality.

And how long before that bans the UK too.

I'm now very happy that I have originals of all my Imgur images in a folder (have done similar for Postimages). That way, my images are always available for alternative file host solutions, should the need arise.

Last edited by phrontister (Yesterday 10:06:24)

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"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson