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We have a circle of certain radius.How many circles of smaller radius(we are provided with the ratio of radius) can be placed within the larger circle? Help me to determine the least number of the smaller circles that can be filled?? I am trying to generate a generalized solution, and looking for help??

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An infinite number of circles may be placed inside the large circle. That is without any restrictions.

You have placed no restriction on whether the smaller circles may or may not intersect each other, may or may not be coplanar, or whether the circles may be or may not be concentric.

Implied in Anza Power's last post, it is assumed the circles may not intersect, may not be coplanar and may not be concentric.

For a clear and precise answer, you would need provide the ratio(s) of the radius, which you state we are provided, but failed to provide.

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An infinite number of circles may be placed inside the large circle. That is without any restrictions.

You have placed no restriction on whether the smaller circles may or may not intersect each other, may or may not be coplanar, or whether the circles may be or may not be concentric.

Implied in Anza Power's last post, it is assumed the circles may not intersect, may not be coplanar and may not be concentric.

For a clear and precise answer, you would need provide the ratio(s) of the radius, which you state we are provided, but failed to provide.

He is looking for a generalized solution.

if you try to form rows

post-31113-017193000 1281974536.jpg

We should be able to calculate the number of circles in the outer row by determining the angle of the smaller circle (see image below). 360/ANGLE = number of circles in the outer row.

post-31113-072655500 1281975856.jpg

I'm not sure if this makes sense but:

The challenging part will then be to determine the number of other rows that are possible, because the circles can recess between the outer circles. At best, you will only be able to fit a circle in every 2nd space if you recess them, however depending on the size of the circle, it may be more optimum to not recces them. I think you would need to write an algorithm:

The steps I would take in writing an algorithm:

1) Determine number of circles in outer row

2) Check each recces position in 1st interior row to see if a circle fits, if at least one fits store as path #1

3) Check the number of circles in 1st interior row without recessing the circles, if at least one fits, store as path #2

4) repeat steps 2 and 3 for next interior row. Try both path #1 and then path #2 (after the 2nd interior row you will have 4 paths). If number of circles possible is 0, end path.

5) Path with highest number of circles is optimum.

I think we could find a solution fairly easily if we didn't recess any of the interior rows of circles - probably even a simple mathematical relationship, however recessing the circles will be a challenge. Recessing the circles will likely be the optimum solution (but may not be for all circle ratios).

Edited by littlej
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LittleJ: if you leave any space between circles, you haven't filled the space with circles, only part of the space. If this were a hexagon filled with hexagons, your tack would be valid.

Also, he asked for the "least" number of circles. What's the smallest number you can think of?

if you try to form rows

post-31113-017193000 1281974536.jpg

We should be able to calculate the number of circles in the outer row by determining the angle of the smaller circle (see image below). 360/ANGLE = number of circles in the outer row.

post-31113-072655500 1281975856.jpg

I'm not sure if this makes sense but:

The challenging part will then be to determine the number of other rows that are possible, because the circles can recess between the outer circles. At best, you will only be able to fit a circle in every 2nd space if you recess them, however depending on the size of the circle, it may be more optimum to not recces them. I think you would need to write an algorithm:

The steps I would take in writing an algorithm:

1) Determine number of circles in outer row

2) Check each recces position in 1st interior row to see if a circle fits, if at least one fits store as path #1

3) Check the number of circles in 1st interior row without recessing the circles, if at least one fits, store as path #2

4) repeat steps 2 and 3 for next interior row. Try both path #1 and then path #2 (after the 2nd interior row you will have 4 paths). If number of circles possible is 0, end path.

5) Path with highest number of circles is optimum.

I think we could find a solution fairly easily if we didn't recess any of the interior rows of circles - probably even a simple mathematical relationship, however recessing the circles will be a challenge. Recessing the circles will likely be the optimum solution (but may not be for all circle ratios).

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LittleJ: if you leave any space between circles, you haven't filled the space with circles, only part of the space. If this were a hexagon filled with hexagons, your tack would be valid.

Also, he asked for the "least" number of circles. What's the smallest number you can think of?

I'm not sure I understand what you are mean - the question is about circles, not hexagons. There is no possible way to fill up all white space. I believe there will be conditions where it is optimum not to put the circles into the recesses. You are limited to 1 circle every two recesses doing this - try it out. You will see the effect as the size of the smaller circles decrease.

A note about the 'least number of circles': I assumed that he just made a typo - it would be fairly obvious that 1 is the least number of circles as long as it is smaller.

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He is looking for a generalized solution.

if you try to form rows

post-31113-017193000 1281974536.jpg

We should be able to calculate the number of circles in the outer row by determining the angle of the smaller circle (see image below). 360/ANGLE = number of circles in the outer row.

post-31113-072655500 1281975856.jpg

I'm not sure if this makes sense but:

The challenging part will then be to determine the number of other rows that are possible, because the circles can recess between the outer circles. At best, you will only be able to fit a circle in every 2nd space if you recess them, however depending on the size of the circle, it may be more optimum to not recces them. I think you would need to write an algorithm:

The steps I would take in writing an algorithm:

1) Determine number of circles in outer row

2) Check each recces position in 1st interior row to see if a circle fits, if at least one fits store as path #1

3) Check the number of circles in 1st interior row without recessing the circles, if at least one fits, store as path #2

4) repeat steps 2 and 3 for next interior row. Try both path #1 and then path #2 (after the 2nd interior row you will have 4 paths). If number of circles possible is 0, end path.

5) Path with highest number of circles is optimum.

I think we could find a solution fairly easily if we didn't recess any of the interior rows of circles - probably even a simple mathematical relationship, however recessing the circles will be a challenge. Recessing the circles will likely be the optimum solution (but may not be for all circle ratios).

From your interpretation of the question, littlej, it sounds as if what is sought is what is called circle packing.

Solutions for the smallest diameter circles into which n unit-diameter circles can be packed have been proved optimal for n = 1 through 10 (Kravitz 1967). The best known results are summarized in the following table,...

n        d exact        d approx.

1	    1	         1.00000

2	    2	         2.00000

3       1 + 2/3•√3       2.15470...

4         1 + √2	 2.41421...

5    1 + √(2•(1+1/√5))   2.70130...

6	    3	         3.00000

7	    3	         3.00000

8	 1 + csc(Ï€/7)    3.30476...

9    1 + √(2•(2 + √2))	 3.61312...

10		         3.82...

11		

12		         4.02...

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He is looking for a generalized solution.

if you try to form rows

post-31113-017193000 1281974536.jpg

We should be able to calculate the number of circles in the outer row by determining the angle of the smaller circle (see image below). 360/ANGLE = number of circles in the outer row.

post-31113-072655500 1281975856.jpg

I'm not sure if this makes sense but:

The challenging part will then be to determine the number of other rows that are possible, because the circles can recess between the outer circles. At best, you will only be able to fit a circle in every 2nd space if you recess them, however depending on the size of the circle, it may be more optimum to not recces them. I think you would need to write an algorithm:

The steps I would take in writing an algorithm:

1) Determine number of circles in outer row

2) Check each recces position in 1st interior row to see if a circle fits, if at least one fits store as path #1

3) Check the number of circles in 1st interior row without recessing the circles, if at least one fits, store as path #2

4) repeat steps 2 and 3 for next interior row. Try both path #1 and then path #2 (after the 2nd interior row you will have 4 paths). If number of circles possible is 0, end path.

5) Path with highest number of circles is optimum.

I think we could find a solution fairly easily if we didn't recess any of the interior rows of circles - probably even a simple mathematical relationship, however recessing the circles will be a challenge. Recessing the circles will likely be the optimum solution (but may not be for all circle ratios).

Thanks for your response.

I have made a slight mistake in stating the problem. I meant to write the arrangement of circles with least gap, but it became something else.

What I have done is something like what u have mentioned littlej. I too determined the no. of circles in the outer row and realized that I may not always get the no . of circles in whole number. The deficit area(due to the fractional part of circle neglected) always seems to cause problem. But the problem seems to analyzed in other way by

1. first placing the smaller circle making t concentric with the larger one.

2. place other circle in its outer circumference, maximum no. of circles being 6, of the same radius.

3. now we can place 6 circles in hexagonal pattern outside each circle. This can be continued till the regular hexagon that can be inscribed in the outer radius is filled. This we can get the maximum compaction between the circles but can't obtain the generalized mathematical solution we require.

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