This just stuck to my mind, and I tried several AI's with no satisfactory answers.
The problem is very short.
How many lines can you draw in a 3D space, which intersect on a single point, and none of them has an angle less than 10° (degrees) to any other? (Angle between any two lines are 10° or more.)
Prove your result.
Thoughts, hints and info:
More hints:
If the problem was about to find 180° (instead of 10°) , the answer is 1 ( overlapping lines cannot be counted).
The problem is very short.
How many lines can you draw in a 3D space, which intersect on a single point, and none of them has an angle less than 10° (degrees) to any other? (Angle between any two lines are 10° or more.)
Prove your result.
Thoughts, hints and info:
- Claude.ai was the best performing among others. It tried a way of steradians. It calculated a vector and its 10° circular area that fills the surface of a containing sphere. By dividing the spheres area to that area the result was simplified to (4𝜋) / ( 2𝜋.(1-cos(5°)) ). The result than becomes 525.58, and it concluded that it must be 525, as you can have only integer number of lines.
- This makes a lot of sense, on the other hand, it somehow accepts the area of the cirle such that, non-euclidian circles fill all the area of a sphere. (which is not true)
- All other AI's failed deeply ( I have not tried Chat GPT-4o)
More hints:
If the problem was about to find 180° (instead of 10°) , the answer is 1 ( overlapping lines cannot be counted).
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