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).