BrainDen.com - Brain Teasers

## Question

1. Simple problem.
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Two unit circles touch each other and a horizontal line segment. What is the radius r of the largest circle that fits in the space bounded by these objects?
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2. Harder problem
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What is the greatest number of circles that can have each pair of circles tangent at a single point?
How many configurations are there?
Can you find a relationship among the radii of the circles?

## Recommended Posts

• 0 r = 1/4

Let O1 be the centre of left circle, O2 the centre of right circle nad O3 the centre of little circle. let the two large circles meet at A.

Then O1O3 = 1+r

O1A = 1

AO3 = 1-r

Since O1AO3 for a right triangle,

(1+r)² = (1-r)² + 1

Simplifying this, we get r = 1/4

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• 0 (R + r)^2 = R^2 + (R - r)^2

R = 4r

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• 0

Could this be the answer for the harder part?

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• 0

Maybe the harder problem requires clarification (for me)... But, if I understood it correctly it's not hard at all

You can fill the entire surface with circles of the same radius in the way that every circle will be touching other circles at a single point like this

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• 0

Maybe the harder problem requires clarification (for me)... It is clearly required for each pair of circles to be tangent, which is not the case in your example. For example the top-left circle is not tangent with bottom-right circle.

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• 0 r = (20.5-1)/2

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• 0 seems to me...

4 is the most. have three circles in a triangular shape, and have the third circle in the middle space.

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• 0 r = (20.5-1)/2

Totally missed it... ##### Share on other sites

• 0

Could this be the answer for the harder part?

Yes, the way the OP is [poorly] worded, that is admissible.

The Hawaiian earring has a common point of tangency, admitting an infinite number of circles.

The OP intended to require the pair-wise points of tangency be distinct.

How many circles can have distinct pairwise tangent points?

How many configurations exist?

Can you find a relationship among the radii?

Bonus question: can you relate this to the first, simple question?

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• 0

seems to me...

4 is the most. have three circles in a triangular shape, and have the third circle in the middle space.

You have it.

Is that the only configuration?

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• 0

The OP intended to require the pair-wise points of tangency be distinct.

The simple part is a special case with one of the circles replaced by a straight line.

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• 0

You have it.

Is that the only configuration?

You can have 3 circles in triangular shape and the 4th surrounding them

The radius of the large surrounding circle is r*(sqrt(3)/2+1) where r is the radius of the 3 other circles.

For the configuration in which a smaller circle is inside the other 3 the radius of the smaller circle is r*(sqrt(3)/2-1)

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• 0

The simple part is a special case with one of the circles replaced by a straight line.

Right. Several years ago I chuckled at a New Yorker cartoon

in which a young boy was tugging on the sleeve of his

father, who was comfortably ensconced in his easy chair.

The caption read: "Go ask your search engine!" ##### Share on other sites

• 0

You can have 3 circles in triangular shape and the 4th surrounding them

The radius of the large surrounding circle is r*(sqrt(3)/2+1) where r is the radius of the 3 other circles.

For the configuration in which a smaller circle is inside the other 3 the radius of the smaller circle is r*(sqrt(3)/2-1)

Yes, that's the other one.

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• 0 Yes, that's the other one.

I suppose we're searching for a proof that this is really the largest number of circles possible in such a configuration (and that there are no other configurations with the same number)? Or is it just that these are the best solutions anyone has ever found?

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• 0

I suppose we're searching for a proof that this is really the largest number of circles possible in such a configuration (and that there are no other configurations with the same number)? Or is it just that these are the best solutions anyone has ever found?

There are two types of tangency: internal and external.

In case when only external tangency exists, 4 is a maximum: we start with 3 circles, then the only option is to put the fourth one "in the middle", and that's it.

Now suppose there are two internally tangent circles L (larger) and S (smaller). Now all other circles have to go inside L (if not, they will not be able to be tangent with S) and have to be externally tangent with S (if not, they will not be able to be tangent with L).

So in this case we have one "big" circle and a number of "small" circles, small circles are pair-wise externally tangent with each other and each small circle is internally tangent with the big circle. In this case at most 3 small circles are possible which is easier "to see", than to prove.

(I'm not suggesting that the prove is hard.)

So 4 is the maximum and only 2 configurations are possible.

PS I believe this reasoning can be extended for 3-dimensional case (mutually touching spheres).

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• 0 There are two types of tangency: internal and external.

In case when only external tangency exists, 4 is a maximum: we start with 3 circles, then the only option is to put the fourth one "in the middle", and that's it.

Now suppose there are two internally tangent circles L (larger) and S (smaller). Now all other circles have to go inside L (if not, they will not be able to be tangent with S) and have to be externally tangent with S (if not, they will not be able to be tangent with L).

So in this case we have one "big" circle and a number of "small" circles, small circles are pair-wise externally tangent with each other and each small circle is internally tangent with the big circle. In this case at most 3 small circles are possible which is easier "to see", than to prove.

(I'm not suggesting that the prove is hard.)

So 4 is the maximum and only 2 configurations are possible.

PS I believe this reasoning can be extended for 3-dimensional case (mutually touching spheres).

It is indeed very easy to "see" as you said, but I'm not convinced the proof is easy. It's one of those things that seem to obvious (and not worthwhile) to try and prove, but would actually be quite difficult. In your explanations for instance, in the case where only external tangents are allowed, it "seems clear" that there is only one option for the fourth sphere, but that is not a topological proof. The internal tangent proof then relies on the external tangent proof (i.e. maybe some other unknown configuration of 4 externally tangent circles could also be tangential to the larger circle, or some other configuration of three - leading to a different configuration of 4).

I agree with you that these are almost certainly the right answers, but that's different from a rigorous proof.

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• 0

I agree with you that these are almost certainly the right answers, but that's different from a rigorous proof.

I totally agree. This is why I wrote "sketch of a proof" instead of "proof".

Besides, I've left a case when there is one large circle (L) and a number of small circles,

small circles are pair-wise externally tangent to each other and each small circle is internally tangent to the big circle.

I just said, that "it is easy to see" that at most 3 small circles are possible, so I just want to complete it.

Suppose we have 4 or more inner circles. Let's number them 1, 2, 3, 4... in the following way:

number 1 is assigned freely, and subsequent numbers are assigned subsequently clockwise according to tangent points to the circle L.

Since circle 2 is tangent to circle 4 and both are tangent to the circle L, they divide interior of circle L in two parts.

But circles 1 and 3 are in different parts, so these circles cannot be tangent.

This contradiction shows that 4 or more small circles are not possible.

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• 0 It is indeed very easy to "see" as you said, but I'm not convinced the proof is easy. It's one of those things that seem to obvious (and not worthwhile) to try and prove, but would actually be quite difficult. In your explanations for instance, in the case where only external tangents are allowed, it "seems clear" that there is only one option for the fourth sphere, but that is not a topological proof. The internal tangent proof then relies on the external tangent proof (i.e. maybe some other unknown configuration of 4 externally tangent circles could also be tangential to the larger circle, or some other configuration of three - leading to a different configuration of 4).

I agree with you that these are almost certainly the right answers, but that's different from a rigorous proof.

Here's a proof:

There are 360° in a circle, in a third of a fourth of a circle there are 30° divide 360 by 30 and you'll get 12, add a 30 to that and you get 42 which is the answer to life, the universe and everything, subtract 2 and divide by 10 and you get 4...

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• 0 Here's a proof:

There are 360° in a circle, in a third of a fourth of a circle there are 30° divide 360 by 30 and you'll get 12, add a 30 to that and you get 42 which is the answer to life, the universe and everything, subtract 2 and divide by 10 and you get 4...

But that proof is mostly harmless. Now if there's a puzzle around here that discusses the question with 42 as the answer, that would be reason to panic...

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• 0 Yes, that's the other one.

Four circles may be the maximum possible where all four circles lie in the same plane, but more than four circles with pair-wise points of tangency can exist in three dimensional space. And even more in poly-dimensional space. Of course, proving this conjecture would be difficult.

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• 0

Four circles may be the maximum possible where all four circles lie in the same plane, but more than four circles with pair-wise points of tangency can exist in three dimensional space. And even more in poly-dimensional space. Of course, proving this conjecture would be difficult.

First, here's a proof that extra dimensions beyond 3D do not buy you anything.

Given two circles that are not coplanar and tangent to each other at point p.

We know that the tangent for both circles at point p must be the intersection of their respective planes.

The vectors from p to the centers of the two circles and the tangent line define a 3D space, S.

Any 2D plane not in S, P1, can intersect with this space with at most 1 line.

For this line to be tangent with both existing circles, it would need to be the tangent line between the two existing circles.

Since we are not allowing the same point for multiple tangencies, no circle on P1 can be tangent to both existing circles.

This is sufficient to show additional dimensions beyond 3D do not allow more circles.

You could have that line touch both circles, but it cannot be tangent to both circles since they are not coplanar

(similarly to show that if the first two circles are coplanar, you can show the third must be as well)

I'm fairly sure I could prove that 4 is the max, but cannot find a simple way to prove it yet.

Then again, if you have an example in 3 space with 5 circles, I'd love to see it.

High dimensional geometry isn't quite in my field of expertise, so if someone sees an error in my proof let me know.

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• 0 For this line to be tangent with both existing circles, it would need to be the tangent line between the two existing circles.

In 2D space, there are at least two tangent lines to any circle. In 3D space, I can visualise an almost infinite number of tangent lines to two non-coplanar circles (though I'm not sure about this). This means that the tangent in the 4th dimension doesn't necessarily have to share a tangent point with the existing circles.

This is not my field of expertise either, so maybe I'm just not getting it.

Edited by rajat_magic
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• 0 In 2D space, there are at least two tangent lines to any circle. In 3D space, I can visualise an almost infinite number of tangent lines to two non-coplanar circles (though I'm not sure about this). This means that the tangent in the 4th dimension doesn't necessarily have to share a tangent point with the existing circles.

This is not my field of expertise either, so maybe I'm just not getting it.

Also, if we allow circles inside other circles, the tangential points don't have to lie on a straight line.

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• 0 Also, if we allow circles inside other circles, the tangential points don't have to lie on a straight line.

Sorry for the third add-on. Was working on my own proof borrowing from your approach, when I realised that we can't actually approach it this way. We're talking about a new circle being tangential to the existing circles, so there is no requirement at all that the two points where the new circle touches the existing circles lie on a tangential line to the existing circles. For example, think of two tangential circles in the 2D space. To add a third tangential circle to this group (still in 2D), the new circle does not touch the existing circles on a tangent line.

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