To minimize the radius ( R ) of the middle circle surrounded by non-overlapping circles of radii ( 1, 2, \ldots, n ), the optimal arrangement involves strategically placing larger circles with smaller ones in between to reduce the sum of products of adjacent radii, thereby minimizing the required angular space and allowing a smaller ( R ).

General Procedure:

1. **Ignore the smallest ra...

Write the radii around the central circle as r1, r2, …, rn in cyclic order, and let R be the radius of the middle circle. If two neighboring outer circles have radii a and b, then the angle at the center between the rays to their centers is θ(a,b;R) = 2 arcsin( sqrt(a/(R+a)) · sqrt(b/(R+b)) ).

Hence, if we define t(r) = sqrt(r/(R+r)) in (0,1), and set ti = t(ri), then for a given cyclic order S(R...

Notation

Let

 r1 , … , rn ( 1 ≤ r1 < r2 < … < rn = n ) (1)

be the radii of the n outer circles and let R be the (unknown) radius of the middle circle.
Write

 αk(R)=arccos(1−2 rk rk+1 ⁄[(R+rk)(R+rk+1)]) ,  k=1,…,n, (rn+1 = r1)

so that the n central angles are α1,…,αn and the closing condition is

 Fπ(P,R):= α1+…+αn = 2π , ...

<think> I need to arrange circles of radii 1 through n around a middle circle so that they're all externally tangent to it and not overlapping, and I want to minimize the radius R of the middle circle. The user has a conjecture about how to do this, but I should verify it or find a better way.

First, for n=10, they have two examples: one with radii in order 1 to 10 giving R≈10.77, and another wit...

Notation
– The radii that have to be placed round the centre are
 1<2<⋯<n.
– We shall write

  θR(x,y)=arccos ( 1− 2xy ⁄[(R+x)(R+y)] )  (1)

 for the angle at the centre that is cut‐off by two adjacent outer circles of radii x and y when the central circle has radius R.

For a concrete ordering σ=(r1,r2,…,rn) of the numbers 1,…,n the condition that the circles close up round the centre ...

Minimizing the Central Circle Radius When Surrounded by Circles of Radii 1 to n

This problem asks how to arrange circles with radii 1 through n around a central circle such that:

1. All surrounding circles are externally tangent to the central circle
2. The surrounding circles don't overlap
3. The radius of the central circle is minimized

Understanding the Mathematical Constraint

The key c...

This is a fascinating optimization problem in discrete geometry. Let me analyze your approach and conjecture.

Analysis of Your Mathematical Formulation

Your formulation is correct. The constraint equation:

is indeed the key constraint, where we want to minimize subject to this constraint and the ...

This is an excellent and well-formulated question about a classic problem in discrete geometry. Your mathematical setup is perfectly correct, and your intuition is heading in the right direction. This problem is known as the "kissing number problem" on a circle, but for circles of unequal radii.

Let's break down the problem to find the optimization strategy.

The Core Idea: Simplifying the Eq...

One can show (by the usual “adjacent‐swap’’ or “pair‐exchange’’ arguments) that in the optimum configuration no two large circles ever sit next to each other – equivalently one wants to keep each big circle “book-ended’’ by smaller ones. In fact the unique (up to rotation or reflection) minimiser is

 “largest, smallest, 2nd–largest, 2nd–smallest, 3rd–largest, 3rd–smallest, …”

or in symbols, if ...