5 ms·

Buffon's needle or how ants estimate nest size

87 points by billconan 7y ago

Rainymood 7y ago
Unreadable on Chrome.
- ChrisClark 7y ago
  Perfectly readable on Chrome.
  - deleted 7y ago
    [deleted]
- billconan 7y ago
  Which platform? Mobile? Which version?
  - Rainymood 7y ago
    Weird. It is fixed now. It got stuck loading something I think and didn't expand beyond the first paragraph. Now it does. My bad.
lorenzhs 7y ago
There’s a nice chapter on Buffon’s needle problem in “Proofs from THE BOOK”, an Erdős-inspired attempt to collect the most beautiful proofs. You can find a PDF of the entire book easily enough on Google, it’s a good read although I would dispute the beauty of some of the proofs in it.
- targonca 7y ago
  >although I would dispute the beauty of some of the proofs in it Agree. In my opinion, the proofs in that book are not so much about beauty, rather how one can prove seemingly complex stuff with surprisingly elementary tools.
runeblaze 7y ago
I thought this was total magic in high school until my college's intro to probability class discussed this before the midterm. I guess maths progression in college can be fast.
- quickthrower2 7y ago
  I think high school is constrained by having to normalize the curriculum over the entire population, but college (I take that to mean university) has more leeway to teach you what they like, and make it harder. Plus you are only studying 1-2 subjects.
will_pseudonym 7y ago
This was a practice problem I remember from when I was studying for Exam 1/P, one of the qualifying exams for actuaries. https://www.soa.org/education/exam-req/edu-exam-p-detail.aspx https://www.soa.org/education/exam-req/edu-exam-p-detail.asp... https://www.casact.org/admissions/syllabus/index.cfm?fa=1syllabi https://www.casact.org/admissions/syllabus/index.cfm?fa=1syl...
malisper 7y ago
An interesting related problem is Bertrand's paradox[0]. You have an equilateral triangle inscribed in a circle. You pick a random chord from the circle (a line that goes from one edge of the circle to another). What's the probability the chord is longer than an edge of the triangle? As it turns out, there isn't a well-defined solution. That's the paradox. The issue comes from "picking a random chord". There are different ways that you can pick a random chord which lead to different solutions. One approach to picking a random chord is to pick two random points on the circle and draw the chord between them. This gives you a probability of 1/3. Another approach is to pick a random direction and radius, then draw a chord at the end of the radius perpendicular to the radius. This approach gives you a probability of 1/2. One interesting aspect is the second approach is the only approach where if you were to draw a circle within the larger circle, the distribution of the chords in the inner circle will match the distribution of chords in the outer circle. [0] https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
- CDSlice 7y ago
  I am not sure what makes this a paradox? Normally I think of a paradox as leading to a state where you have two or more contradictory statements being true at like Russell's Paradox in set theory. This just seems like a problem where you don't have enough information to solve it but once you get the info you need (how to pick random chords) you get a logically consistent solution.
  - malisper 7y ago
    A paradox is any statement that appears to contradict itself. That includes both actual contradictions such as Russel's paradox and statements that appear to be invalid, but have non-obvious valid conclusions like the Monty Hall problem. This problems falls into the second category. We are given what appears to be a well-defined problem, but when we solve it in different ways, we get different answers. The resolution to the paradox is the problem isn't well defined because your technique for generating randomness will produce different answers.
samsamoa 7y ago
A beautiful related result is the Crofton Formula [0], which says that you can measure the length of any curve by appropriately counting the number of straight lines that intersect it. (My co-authors and I discovered a generalization of the formula that also holds in curved space [1].) [0] https://en.wikipedia.org/wiki/Crofton_formula https://en.wikipedia.org/wiki/Crofton_formula [1] https://arxiv.org/pdf/1505.05515.pdf https://arxiv.org/pdf/1505.05515.pdf
- sova 7y ago
  I'm especially intrigued by the "Timelike, Spacelike, and Lightlike" diagram on Page 10 of the linked paper where "lightlike" phenomena share an endpoint. Fascinating.
- srean 7y ago
  Interesting as the paper is, please link to the arxiv abstract rather than the pdf as that's more web friendly. Reader can always click on he pdf link there. https://arxiv.org/abs/1505.05515 https://arxiv.org/abs/1505.05515
deleted 7y ago
[deleted]
safgasCVS 7y ago
I first saw this little gem in a stats test in my second year at university. I must say it was a lot more fun deriving it without the time pressure
lonelappde 7y ago
Kakeya is another nice needle problem: https://youtu.be/IM-n9c-ARHU https://youtu.be/IM-n9c-ARHU
deleted 7y ago
[deleted]

87 points by billconan 7y ago

Rainymood 7y ago

Unreadable on Chrome.

ChrisClark 7y ago
Perfectly readable on Chrome.
- deleted 7y ago
  [deleted]
billconan 7y ago
Which platform? Mobile? Which version?
- Rainymood 7y ago
  Weird. It is fixed now. It got stuck loading something I think and didn't expand beyond the first paragraph. Now it does. My bad.

lorenzhs 7y ago

There’s a nice chapter on Buffon’s needle problem in “Proofs from THE BOOK”, an Erdős-inspired attempt to collect the most beautiful proofs. You can find a PDF of the entire book easily enough on Google, it’s a good read although I would dispute the beauty of some of the proofs in it.

targonca 7y ago
>although I would dispute the beauty of some of the proofs in it Agree. In my opinion, the proofs in that book are not so much about beauty, rather how one can prove seemingly complex stuff with surprisingly elementary tools.

runeblaze 7y ago

I thought this was total magic in high school until my college's intro to probability class discussed this before the midterm. I guess maths progression in college can be fast.

quickthrower2 7y ago
I think high school is constrained by having to normalize the curriculum over the entire population, but college (I take that to mean university) has more leeway to teach you what they like, and make it harder. Plus you are only studying 1-2 subjects.

will_pseudonym 7y ago

This was a practice problem I remember from when I was studying for Exam 1/P, one of the qualifying exams for actuaries. https://www.soa.org/education/exam-req/edu-exam-p-detail.aspx https://www.soa.org/education/exam-req/edu-exam-p-detail.asp... https://www.casact.org/admissions/syllabus/index.cfm?fa=1syllabi https://www.casact.org/admissions/syllabus/index.cfm?fa=1syl...

malisper 7y ago

An interesting related problem is Bertrand's paradox[0]. You have an equilateral triangle inscribed in a circle. You pick a random chord from the circle (a line that goes from one edge of the circle to another). What's the probability the chord is longer than an edge of the triangle? As it turns out, there isn't a well-defined solution. That's the paradox. The issue comes from "picking a random chord". There are different ways that you can pick a random chord which lead to different solutions. One approach to picking a random chord is to pick two random points on the circle and draw the chord between them. This gives you a probability of 1/3. Another approach is to pick a random direction and radius, then draw a chord at the end of the radius perpendicular to the radius. This approach gives you a probability of 1/2. One interesting aspect is the second approach is the only approach where if you were to draw a circle within the larger circle, the distribution of the chords in the inner circle will match the distribution of chords in the outer circle. [0] https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

CDSlice 7y ago
I am not sure what makes this a paradox? Normally I think of a paradox as leading to a state where you have two or more contradictory statements being true at like Russell's Paradox in set theory. This just seems like a problem where you don't have enough information to solve it but once you get the info you need (how to pick random chords) you get a logically consistent solution.
- malisper 7y ago
  A paradox is any statement that appears to contradict itself. That includes both actual contradictions such as Russel's paradox and statements that appear to be invalid, but have non-obvious valid conclusions like the Monty Hall problem. This problems falls into the second category. We are given what appears to be a well-defined problem, but when we solve it in different ways, we get different answers. The resolution to the paradox is the problem isn't well defined because your technique for generating randomness will produce different answers.

samsamoa 7y ago

A beautiful related result is the Crofton Formula [0], which says that you can measure the length of any curve by appropriately counting the number of straight lines that intersect it. (My co-authors and I discovered a generalization of the formula that also holds in curved space [1].) [0] https://en.wikipedia.org/wiki/Crofton_formula https://en.wikipedia.org/wiki/Crofton_formula [1] https://arxiv.org/pdf/1505.05515.pdf https://arxiv.org/pdf/1505.05515.pdf

sova 7y ago
I'm especially intrigued by the "Timelike, Spacelike, and Lightlike" diagram on Page 10 of the linked paper where "lightlike" phenomena share an endpoint. Fascinating.
srean 7y ago
Interesting as the paper is, please link to the arxiv abstract rather than the pdf as that's more web friendly. Reader can always click on he pdf link there. https://arxiv.org/abs/1505.05515 https://arxiv.org/abs/1505.05515

deleted 7y ago

[deleted]

safgasCVS 7y ago

I first saw this little gem in a stats test in my second year at university. I must say it was a lot more fun deriving it without the time pressure

lonelappde 7y ago

Kakeya is another nice needle problem: https://youtu.be/IM-n9c-ARHU https://youtu.be/IM-n9c-ARHU

deleted 7y ago

[deleted]