First post, any help would be greatly appreciated

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• Let me start by saying I m far from a mathematician, however this question was posed to me many years ago in high school and I was never able to find a
Message 1 of 7 , Aug 7, 2003
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Let me start by saying I'm far from a mathematician, however this
question was posed to me many years ago in high school and I was never
able to find a solution, furthermore I've asked over 30 people with
widely varying math experience and none have been able to answer it, and
to top it off I once physically modeled over 30 iterations of this
problem using coins of varying denominations, measuring with a ruler and
finally fed the resultant answer table into a genetic algorithm based
program which could not generate better than a 90~% fit for the answer
givens it.

So Here Goes:

Given an equilateral triangle with a base of X, what is the maximum
number(Z) of circles of radius Y that can be inscribed within the
boundaries of the triangle?

An answer to this one would totally rock.

TIA

-Dave Pratt
Berkeley, CA

[Non-text portions of this message have been removed]
• ... Either I don t understand the question or it is trivial: The largest circle you can inscribe in an equilateral triangle is the incircle. Call its radius r.
Message 2 of 7 , Aug 7, 2003
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>
> Given an equilateral triangle with a base of X, what is the maximum
> number(Z) of circles of radius Y that can be inscribed within the
> boundaries of the triangle?
>

Either I don't understand the question or it is trivial:

The largest circle you can inscribe in an equilateral triangle is
the incircle. Call its radius r. So for the above question,

If Y>r then Z is zero
If Y=r then Z is one
If Y < r then Z is infinity.
• ... I think he means: how many circles can be inscribed simultaneously without overlapping each other? I don t know the answer, but it must be an interesting
Message 3 of 7 , Aug 7, 2003
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In Hyacinthos message #7445, Michael Lambrou wrote:

>> > Given an equilateral triangle with a base of X, what
>> > is the maximum number(Z) of circles of radius Y that
>> > can be inscribed within the boundaries of the
>> > triangle?
>> >
>>
>> Either I don't understand the question or it is trivial:

I think he means: how many circles can be inscribed
simultaneously without overlapping each other?

I don't know the answer, but it must be an interesting
question!

Darij Grinberg
• Darij, your interpretation of the question is correct. -Dave ... From: Darij Grinberg [mailto:darij_grinberg@web.de] Sent: Thursday, August 07, 2003 7:55 AM
Message 4 of 7 , Aug 7, 2003
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Darij, your interpretation of the question is correct.

-Dave

-----Original Message-----
From: Darij Grinberg [mailto:darij_grinberg@...]
Sent: Thursday, August 07, 2003 7:55 AM
To: Hyacinthos@yahoogroups.com
Subject: [ipod] RE: [EMHL] First post, any help would be greatly
appreciated

In Hyacinthos message #7445, Michael Lambrou wrote:

>> > Given an equilateral triangle with a base of X, what
>> > is the maximum number(Z) of circles of radius Y that
>> > can be inscribed within the boundaries of the
>> > triangle?
>> >
>>
>> Either I don't understand the question or it is trivial:

I think he means: how many circles can be inscribed
simultaneously without overlapping each other?

I don't know the answer, but it must be an interesting
question!

Darij Grinberg

[Non-text portions of this message have been removed]
• Dear all, [Dave] ... [Michael] ... [Darij] ... ******* If Darij, is right then I think that if we could put n of these circles on the base of the triangle then
Message 5 of 7 , Aug 7, 2003
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Dear all,

[Dave]
>>> > Given an equilateral triangle with a base of X, what
>>> > is the maximum number(Z) of circles of radius Y that
>>> > can be inscribed within the boundaries of the
>>> > triangle?

[Michael]
>>> Either I don't understand the question or it is trivial:

[Darij]
>I think he means: how many circles can be inscribed
>simultaneously without overlapping each other?
>
>I don't know the answer, but it must be an interesting
>question!

*******
If Darij, is right then I think that if we could put
n of these circles on the base of the triangle then
2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
n = [(X + 2Y - 2Y*sqrt(3))/2Y]
where [ ] denotes integer part.
Over these circles we could put n - 1 circles and so on.
Hence the total maximum number of circles is Z = n(n+1)/2
and n the number given above.

Best regards
• Wow Nikolaos, that was a very fast response. However, the last math class I took was about 5 years ago and it was a very basic high school level geometry
Message 6 of 7 , Aug 7, 2003
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Wow Nikolaos, that was a very fast response. However, the last math
class I took was about 5 years ago and it was a very basic high school
level geometry class. Do you think you could walk me through your
solution with a little more detail?

-Dave

-----Original Message-----
Sent: Thursday, August 07, 2003 11:42 AM
To: Hyacinthos@yahoogroups.com
Subject: [mathdorks] Re: [EMHL] First post, any help would be greatly
appreciated

Dear all,

[Dave]
>>> > Given an equilateral triangle with a base of X, what
>>> > is the maximum number(Z) of circles of radius Y that
>>> > can be inscribed within the boundaries of the
>>> > triangle?

*******
If Darij, is right then I think that if we could put
n of these circles on the base of the triangle then
2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
n = [(X + 2Y - 2Y*sqrt(3))/2Y]
where [ ] denotes integer part.
Over these circles we could put n - 1 circles and so on.
Hence the total maximum number of circles is Z = n(n+1)/2
and n the number given above.

Best regards
• I don t think it s quite as simple. There s a fairly large literature on this packing problem. If n is large and you can t quite get n circles in the
Message 7 of 7 , Aug 7, 2003
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I don't think it's quite as simple. There's a fairly
large literature on this packing problem.

If n is large and you can't quite get n circles
in the bottom row, then you can get more than

(n-1) + (n-2) + ... + 2 + 1

by spacing the bottom row so that the 2nd, 3rd, ...
rows don't need to be quite so high. Eventually,
there'll be room for more circles at the top.

See, for example R L Graham & B D Lubachevsky, Dense
packings of equal disks in an equilateral triangle:
from 22 to 34 and beyond, {\it Electron. J. Combin.},
{\bf2}(1995) Article 1, 39pp; MR 96a:52027. R.

On Thu, 7 Aug 2003, Nikolaos Dergiades wrote:

> Dear all,
>
> [Dave]
> >>> > Given an equilateral triangle with a base of X, what
> >>> > is the maximum number(Z) of circles of radius Y that
> >>> > can be inscribed within the boundaries of the
> >>> > triangle?
>
>
> [Michael]
> >>> Either I don't understand the question or it is trivial:
>
> [Darij]
> >I think he means: how many circles can be inscribed
> >simultaneously without overlapping each other?
> >
> >I don't know the answer, but it must be an interesting
> >question!
>
>
> *******
> If Darij, is right then I think that if we could put
> n of these circles on the base of the triangle then
> 2Y(n -1)+2Y*sqrt(3) <= X < 2Yn+2Y*sqrt(3) or that
> n = [(X + 2Y - 2Y*sqrt(3))/2Y]
> where [ ] denotes integer part.
> Over these circles we could put n - 1 circles and so on.
> Hence the total maximum number of circles is Z = n(n+1)/2
> and n the number given above.
>
> Best regards