5 ms·

I always liked this problem... What packs more efficiently in a barrel; tennis balls, marbles, or a mixture of tennis balls and marbles? It feels like the sma

I always liked this problem... What packs more efficiently in a barrel; tennis balls, marbles, or a mixture of tennis balls and marbles? It feels like the smaller marbles are denser but obviously they actually pack the same efficiency as the tennis balls or any other sphere, the mixture packs more efficiently.

Kednicma 6y ago
Yes. I really like the metallurgy version: An alloy can be denser than pure elements.
- jld 6y ago
  I was just talking to a friend about how 1 part water plus 1 part ethanol becomes 1.92 parts solution. This raised a question for me that I have yet to research/answer. Maybe one of you knows... if the above solution is 50% ABV, what happens if I add one more part alcohol to the solution? Is it now 66.6% ABV? More? Less? How does ABV take into account the fact that this solution is packed together tighter than its constituent parts?
  - tedunangst 6y ago
    The wikipedia article has all the charts and formulas you need. https://en.wikipedia.org/wiki/Alcohol_by_volume https://en.wikipedia.org/wiki/Alcohol_by_volume (To make the above solution 50% abv, btw, you need to keep adding water until it's 2 (liters, whatever) total volume.)
  - tkeAmarktinClss 6y ago
    Thank you for being the best of HN. The muderation might be awful, but the users are sharp. Learned something new, and I'm a chem grad.
  - riffraff 6y ago
    This is a common issue for people doing liquors at home: you mix 1l of alcohol which you used to extract some flavour with 1l of water and don't get back 2l.
endtime 6y ago
The mixture seems like the intuitive answer to me. A single tennis ball packs the volume of a tennis ball more efficiently than the equivalent volume of marbles.
- gowld 6y ago
  The problem is ambiguously stated. A "volume of X" isn't well-defined when the point of the problem is to reason about packing efficiency.
kens 6y ago
> It feels like the smaller marbles are denser but obviously they actually pack the same efficiency as the tennis balls That's not obvious to me. Won't the marbles pack more efficiently due to the edges of the container? As a limiting case, consider a container slightly smaller than 1 tennis ball: the packing efficiency of tennis balls will be 0%, while marbles will be something like 60-70%.
- dmurray 6y ago
  A "barrel" though is way larger than a tennis ball, so we're not close to the limiting case.
- Retric 6y ago
  Packing efficiency is usually defined for an infinite space because finite containers don’t always favor the same size object. Consider a marble with a diameter of 2inch and a tennis ball with diameter if 3 inches. A cube of length 9.01 inches now favors the tennis balls where one of 8.01 inches favors the marbles.
  - OscarCunningham 6y ago
    According to http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html a cube of side 9.01 can fit 27 balls of diameter 3, but 100 balls of diameter 2. Since 27×3^3 = 729 and 8×2^3 = 800, the balls of diameter 2 are more efficient in this case.
    - OscarCunningham 6y ago
      In fact it seems that the diameter 3 balls are most efficient only for cubes of edge-length between 3 and 2+sqrt(2) = 3.414.... These are the sizes for which you can fit in 1 diameter 3 ball, but can't fit in 4 diameter 2 balls. Above that, diameter 2 balls are more efficient (except that a cube of edge-length 6 can fit in the same volume either way; 27 diameter 2 balls or 8 diameter 3 balls).
    - Retric 6y ago
nanidin 6y ago
Define efficiency - marbles are more dense and will fill the barrel with more mass.
- deleted 6y ago
  [deleted]
- ponker 6y ago
  https://en.m.wikipedia.org/wiki/Packing_density https://en.m.wikipedia.org/wiki/Packing_density
missblit 6y ago
Assuming the barrel is an infinite space, and the marbles are small enough to fit in the gaps from an optimal tennis-ball sphere packing: Not just tennis balls, because after adding all the tennis balls you can, there's still room for marbles to fill in the gaps Not just marbles, because the optimal sphere packing density is the same whether it's all marbles or all tennis balls. Since all tennis balls wasn't optimal, neither is all marbles. I figure things get interesting once the ratio between the two spheres gets closer to 1. (And checking my work, this argument is described in a fancier sciencey way by https://en.wikipedia.org/wiki/Sphere_packing#Unequal_sphere_packing https://en.wikipedia.org/wiki/Sphere_packing#Unequal_sphere_... )

Kednicma 6y ago

Yes. I really like the metallurgy version: An alloy can be denser than pure elements.

jld 6y ago
I was just talking to a friend about how 1 part water plus 1 part ethanol becomes 1.92 parts solution. This raised a question for me that I have yet to research/answer. Maybe one of you knows... if the above solution is 50% ABV, what happens if I add one more part alcohol to the solution? Is it now 66.6% ABV? More? Less? How does ABV take into account the fact that this solution is packed together tighter than its constituent parts?
- tedunangst 6y ago
  The wikipedia article has all the charts and formulas you need. https://en.wikipedia.org/wiki/Alcohol_by_volume https://en.wikipedia.org/wiki/Alcohol_by_volume (To make the above solution 50% abv, btw, you need to keep adding water until it's 2 (liters, whatever) total volume.)
- tkeAmarktinClss 6y ago
  Thank you for being the best of HN. The muderation might be awful, but the users are sharp. Learned something new, and I'm a chem grad.
- riffraff 6y ago
  This is a common issue for people doing liquors at home: you mix 1l of alcohol which you used to extract some flavour with 1l of water and don't get back 2l.

endtime 6y ago

The mixture seems like the intuitive answer to me. A single tennis ball packs the volume of a tennis ball more efficiently than the equivalent volume of marbles.

gowld 6y ago
The problem is ambiguously stated. A "volume of X" isn't well-defined when the point of the problem is to reason about packing efficiency.

kens 6y ago

> It feels like the smaller marbles are denser but obviously they actually pack the same efficiency as the tennis balls That's not obvious to me. Won't the marbles pack more efficiently due to the edges of the container? As a limiting case, consider a container slightly smaller than 1 tennis ball: the packing efficiency of tennis balls will be 0%, while marbles will be something like 60-70%.

dmurray 6y ago
A "barrel" though is way larger than a tennis ball, so we're not close to the limiting case.
Retric 6y ago
Packing efficiency is usually defined for an infinite space because finite containers don’t always favor the same size object. Consider a marble with a diameter of 2inch and a tennis ball with diameter if 3 inches. A cube of length 9.01 inches now favors the tennis balls where one of 8.01 inches favors the marbles.
- OscarCunningham 6y ago
  According to http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html http://hydra.nat.uni-magdeburg.de/packing/scu/scu.html a cube of side 9.01 can fit 27 balls of diameter 3, but 100 balls of diameter 2. Since 27×3^3 = 729 and 8×2^3 = 800, the balls of diameter 2 are more efficient in this case.
  - OscarCunningham 6y ago
    In fact it seems that the diameter 3 balls are most efficient only for cubes of edge-length between 3 and 2+sqrt(2) = 3.414.... These are the sizes for which you can fit in 1 diameter 3 ball, but can't fit in 4 diameter 2 balls. Above that, diameter 2 balls are more efficient (except that a cube of edge-length 6 can fit in the same volume either way; 27 diameter 2 balls or 8 diameter 3 balls).
  - Retric 6y ago

nanidin 6y ago

Define efficiency - marbles are more dense and will fill the barrel with more mass.

deleted 6y ago
[deleted]
ponker 6y ago
https://en.m.wikipedia.org/wiki/Packing_density https://en.m.wikipedia.org/wiki/Packing_density

missblit 6y ago

Assuming the barrel is an infinite space, and the marbles are small enough to fit in the gaps from an optimal tennis-ball sphere packing: Not just tennis balls, because after adding all the tennis balls you can, there's still room for marbles to fill in the gaps Not just marbles, because the optimal sphere packing density is the same whether it's all marbles or all tennis balls. Since all tennis balls wasn't optimal, neither is all marbles. I figure things get interesting once the ratio between the two spheres gets closer to 1. (And checking my work, this argument is described in a fancier sciencey way by https://en.wikipedia.org/wiki/Sphere_packing#Unequal_sphere_packing https://en.wikipedia.org/wiki/Sphere_packing#Unequal_sphere_... )