What is the least number of smaller circles that can be fitted inside a mother circle under certain conditions?

[Munim]

PROBLEM STATEMENT:

What is the least number of smaller circles that can be fitted inside a mother circle under the following conditions:

1.       The smaller circles cannot intersect or be contained inside any other circle besides the mother circle.

2.     The areas of the smaller circles must be N (N<1) times any existing circle inside the mother circle.

3.       The system must contain the maximum number of circles of the same area as possible.

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MUNIM'S PROBLEM.docx

[Genady]

Can you solve it for line segments instead of circles? I.e.,

What is the least number of smaller segments that can be fitted inside a mother segment under the following conditions:

1. The smaller segments cannot intersect or be contained inside any other segment besides the mother segment.

2. The lengths of the smaller segments must be N (N<1) times any existing segment inside the mother segment.

3. The system must contain the maximum number of segments of the same length as possible.

[Munim]

It will be a relatively simple solution for segments as I can vaguely visualize that but not for circles. 

[Munim]

PROBLEM STATEMENT:

What is the least number of circles C_i that can be fitted inside a circle C_M under the following conditions i.e. solve for the least value of n for a certain (L, i):

1. The circles C_i cannot share a common area.

2. The areas of the circles C_i must be A_i=Lⁱ*A_M (where, L<1 & i=1,2,3,...n; A_M=area of circle C_M).

3. The system must contain the maximum number of circles C_i of the same area as possible.

A representation of the problem is attached to the post.

NB: It's an alternate statement to my previous post

[Genady]

I don't understand the new problem statement, specifically, the least value of n for a certain i, where i=1,2,3,...n.

[studiot]

Where did this problem come from please ?

Is it coursework ?

 

Also you should not post duplicate threads, even if you second picture is prettier than the first.

 

It is a form of generalised Malfatti Problem in computational geometry, with the outer boundary being a circle not a triangle.

[MigL]

Is the problem stated correctly, or am I fairly obtuse today.

I see contradictory requirements, such as
"the least number of smaller circles that can be fitted inside a mother circle"
and
"must contain the maximum number of circles of the same area"

If not for the second requirement ( above ) the answer would be trivially simple.
One circle with the radius approaching the limit of the mother circle.

What am I missing ?

[CharonY]

!

Moderator Note

Similar topics merged.

 

[Munim]

19 hours ago, Genady said:

I don't understand the new problem statement, specifically, the least value of n for a certain i, where i=1,2,3,...n.

The problem is to solve for the least number of circles C_i that can be drawn inside the circle C_M. So, we need the least number of circles eg. C₁,C₂,C₃,...,C8 ( as shown in the attached image of the problem) i.e. the value of n=8 for a certain value of L (L<1).

18 hours ago, MigL said:

Is the problem stated correctly, or am I fairly obtuse today.

I see contradictory requirements, such as
"the least number of smaller circles that can be fitted inside a mother circle"
and
"must contain the maximum number of circles of the same area"

If not for the second requirement ( above ) the answer would be trivially simple.
One circle with the radius approaching the limit of the mother circle.

What am I missing ?

Following the image attached to the post, we are looking for the least number of circles C_i, that is, in the image we had to use up to C₈, so, the total n=8, which is the supposed least value of i for a certain value of L. Furthermore, we had to use C₁ once and C₂ also once because we cannot fit more than one of them each, BUT, we can fit two of C₃, C4, C₅ and C₆ and ten C₇ circles. By the condition stated we cannot draw one C3, C₄, C_5, C₆ nor can we draw less than ten C₇. We have to maximize the number of certain size of circles until there is not enough space left to fit that particular size of circle. If we run out of space for C₇, we start drawing C₈ ( the smaller blue button sized circles) and eventually if we run out of space for C₈, we have to try fitting C₉=L⁹.A_M. But if the space left after fitting circles up to C₈ is not enough to fit even one C₉ circle, then we have reached the end of solution for that value of L.

19 hours ago, studiot said:

Where did this problem come from please ?

Is it coursework ?

 

Also you should not post duplicate threads, even if you second picture is prettier than the first.

 

It is a form of generalised Malfatti Problem in computational geometry, with the outer boundary being a circle not a triangle.

It is not any problem I have ever seen anywhere else. It is an unique problem I thought of. You can easily visualize this problem for values suppose L=0.5, because there is just one circle that can be fitted (there is a limiting value for L<0.5 for which there is just one solution of n, which is another problem to solve) as the next circle C₂=0.5*A_M is impossible to be fitted inside C_M. But as L moves on towards zero, the value of n increases.

[Munim]

The attached is an another way to clarify the problem