Math Is Fun Forum

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,818

Archimedes' cattle problem

I just saw Bobby's reference in Jane's "Pell's equation" thread to the "cattle of the sun" problem, which reminded me that I'd recently been introduced to it. It's commonly known as Archimedes' cattle problem, and much has been written about it.

The problem is in two parts: one that I managed to do and one that I didn't .  I enjoyed solving part 1, for which only high school maths is required. Part 2 requires something more advanced - like Pell's equation, for instance.

This link - http://www.cs.drexel.edu/~crorres/Archimedes/Cattle/Statement.html - contains the entire puzzle (in old poetic-style language) and the solution to both parts (for which Maple was used), but I'll post just the easier part in simple form:

The sun god had a herd of cattle consisting of bulls and cows, of which some were white, some black, some yellow and the rest dappled.

From among the bulls:
* the number of the white was one half plus one third the number of the black, plus the yellow
* the number of the black was one quarter plus one fifth the number of the dappled, plus the yellow
* the number of the dappled was one sixth plus one seventh the number of the white, plus the yellow.

From among the cows:
* the number of the white was one third plus one quarter the total of the black cattle
* the number of the black was one quarter plus one fifth the total of the dappled cattle
* the number of the dappled was one fifth plus one sixth the total of the yellow cattle
* the number of the yellow was one sixth plus one seventh the total of the white cattle.

What was the cattle composition for the smallest-possible herd?

The last half of the question isn't in the original, but I saw it somewhere and thought it was a good addition to obtain a unique solution - otherwise there are multiple solutions. I think I read that solving part 2 restricts part 1's solution to the smallest one, but I haven't tested that as I didn't solve part 2.

Last edited by phrontister (2009-08-10 03:30:00)

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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: Archimedes' cattle problem

Hi phrontister;

There are 2 articles one is by Ilan Vardi. He shows how to generate the smallest solution and all the rest. Just remember it is a 207,000 digit number.

Last edited by bobbym (2009-08-10 15:24:11)

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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: 4,818

Re: Archimedes' cattle problem

Thanks, Bobby.

I downloaded Vardi's article. Most of the internet links I could find were to .ps files, for which it seems I need to install a reader my computer doesn't have. I found a .pdf file at:-
http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=35990630E61962A0BE5D8EDD171CB9B1?doi=10.1.1.33.4288&rep=rep1&type=pdf

I read the article, but the explanation to the part 2 solution, in particular, is way beyond my grasp and confirms that I'm doomed to limiting my experience with the cattle problem to part 1.

For part 1 Vardi mentions using Mathematica and gives the final equations for use with that program's Solve command. Again - too advanced for me.

My solution to part 1 is a simple, step-by-step algebraic one...which I'll post when I've set it out sufficiently well. Maybe others might like to try this non-computer approach in the meantime.

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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: Archimedes' cattle problem

Hi phrontister;

ps. files require ghost script to view. You can view them with almost any linux distro.

  I haven't looked at that pdf yet but I know what you are talking about with part 2. Vardi is way over my head on a lot of it. Post what you do understand. Also are you familiar with wolfram alpha? Wolfram has made Mathematica available online. This is a tremendous resource that few people know about.

Last edited by bobbym (2009-08-10 15:26:09)

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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: 4,818

Re: Archimedes' cattle problem

Hi Bobby,

ps. files require ghost script to view. You can view them with almost any linux distro.

Yes - after I downloaded the .pdf file I installed GhostScript, and it read the .ps file. I prefer the .pdf.

I don't have linux.

I didn't know about Wolfram Alpha or online Mathematica...I'll have a look. Thanks.

Last edited by phrontister (2009-08-10 15:53:41)

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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: Archimedes' cattle problem

Hi phrontister;

Oh no, I didn't mean for you to install that piece of junk, ghost script. I prefer .djvu to either .pdf or .ps

I use a puppy linux on a cd or usb to read the occasional ps. Puppy will then change them over to a pdf.

---

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: 4,818

Re: Archimedes' cattle problem

I'd already uninstalled GhostScript, Bobby, as I nearly never have to read .ps files. I can't see that I'd have much use for djvu or puppy, either.

I use PDF-XChange Viewer for pdf, as it has a few annotation tools that I find handy.

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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: Archimedes' cattle problem

I use PDF-XChange Viewer also. .djvu has been threatening to replace the .pdf and .jpg formats for several years now.

---

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: 4,818

Re: Archimedes' cattle problem

Hi Bobby,

Post what you do understand.

I've set it out as best I can. My explanation is a bit amateurish, but it's step-by-step how I went about solving it, using algebra and substitutions.

Edit: Ignore the image below. It's been replaced by the one in the first hide box above, but I can't delete the original.

Last edited by phrontister (2017-02-27 11:29:32)

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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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Archimedes' cattle problem

Hi phrontister;

Looks like you understood the Vardi paper to me.

---

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: 4,818

Re: Archimedes' cattle problem

I only understand Part 1, Bobby, which is just a simple algebraic solution.

Part 2's solution uses advanced techniques that I have no idea about whatsoever, and I find Vardi's language and the concepts he presents just going straight over the top of my head. I think that trying to come to grips with it all would be very difficult for me, given my limited maths knowledge.

Last edited by phrontister (2009-08-12 11:02:13)

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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: Archimedes' cattle problem

Hi phrontister;

  Sorry, I was vague there. I meant for the first part.

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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.