How many tubes in a cylinder - circles in a circle

SwingsitlikeHogan

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The question is causing windmills in my mind 😘

Round, like a circle in a spiral
Like a wheel within a wheel
Never ending or beginning
On an ever-spinning reel
Like a snowball down a mountain
Or a carnival balloon
Like a carousel that's burning
Running rings around the moon

Like a clock whose hands are sweeping
Past the minutes of its face
And the world is like an apple
Whirling silently in space
Like the circles that you find
In the windmills of your mind.
 

backwoodsman

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Chat GPT says 19 for the 71mm and 37 for the 102.

To determine how many 13 mm diameter circles can fit inside a 71 mm diameter circle using hexagonal packing, we need to follow a systematic approach:

1. **Calculate the radius of both circles**:
- Radius of the large circle, \( R \) = 71 mm / 2 = 35.5 mm
- Radius of the small circle, \( r \) = 13 mm / 2 = 6.5 mm

2. **Arrange circles in layers**:
- The distance from the center of one small circle to the center of an adjacent small circle is \( 2r \) = 13 mm.
- Circles are arranged in a hexagonal pattern around a central circle.

3. **Determine the number of layers**:
- The distance from the center of the large circle to the center of circles in the \( n \)-th layer is approximately \( n \times 2r \).
- Calculate the maximum number of layers (\( n \)) that fit within the radius \( R \):
\[
R = n \times 2r \implies 35.5 = n \times 13 \implies n \approx 2.73
\]
- So, there can be 2 complete layers around the central circle.

4. **Count the circles**:
- **Layer 0**: 1 circle (center)
- **Layer 1**: 6 circles (1st layer around the center)
- **Layer 2**: 12 circles (2nd layer around the center)

So, let's verify the arrangement and total count of circles:

1. **Layer 0** (center): 1 circle
2. **Layer 1** (around the center): 6 circles
3. **Layer 2** (around the first layer): 12 circles

Adding them together:
\[
1 + 6 + 12 = 19 \text{ circles}
\]

Thus, 19 circles of 13 mm diameter can fit inside a 71 mm diameter circle using efficient hexagonal packing.
And therein, lies the problem with artificial intelligence.
 

SocketRocket

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The question is causing windmills in my mind 😘

Round, like a circle in a spiral
Like a wheel within a wheel
Never ending or beginning
On an ever-spinning reel
Like a snowball down a mountain
Or a carnival balloon
Like a carousel that's burning
Running rings around the moon

Like a clock whose hands are sweeping
Past the minutes of its face
And the world is like an apple
Whirling silently in space
Like the circles that you find
In the windmills of your mind.
And they're coming to take me away ha haaa
They're coming to take me away ho ho hee hee ha haaa
To the happy home with trees and flowers and chirping birds
And basket weavers who sit and smile and twiddle their thumbs and toes
And they're coming to take me away ha haaa
 
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