Is there a significant difference?
Let’s say I have a container that has neatly stacked coins in it, and the column of coins are all as close together as possible. The stacks come up exactly to the top of the container.
Then, I empty them into a bag, and pour the contents back into the original container.
Does the volume of coins change (much), and in which direction? Space left over? Overflow?
I could do this experimentally, but thought there might be a quicker answer.
I think more information is required: The size and shape of the container and of the coins for a start. Clearly - if the container were one of those tubes that hold ten quarters, then that will be true, whatever way you drop them in unless one or more stands up on its edge.
logic and experience says that neatly stacked will always be more efficient.
Eh. . . OK. How about circular, diameter 190.5 mm, height 152 mm.
Let’s assume that the container is large enough that edge effects are insignificant.
I’d certainly expect that neatly stacked would be the most efficient, but I have no idea what the actual ratio would be.
Numerical simulations in this article describe the random packing densities of basic 3D objects, including cylinders. The best random packing is far from optimal, and coin-shaped cylinders are far from the optimum aspect ratio for cylinders. It seems that when you dump them back in you could get 20% overflow.
ETA the optimal density is nearly 91%
Sorry, ignore the 20%; you would get some overflow but not that much,
My guesstimate is that the optimal 90% density would decrease to something like 80%; it would be interesting if someone ran a few experiments.
I’ll get a chance in a few months when I open my coin bank. And when I do, I’ll be sure to post results.
It may be difficult to define “random” precisely. If you start with a container filled “randomly” and shake it, the coins will arrange themselves into spherical stacks. If the ratio of container and coin diameters is small, these stacks formed by shaking might outperform regular stacks: stack segments can be positioned at angles, and the stack axis need not be perpendicular to its coins.
Even gentle shaking while filling the container will increase packing density.
Good point. Theoretically, if you shook the container vigorously enough, and then decreased the shaking gradually enough, you should ultimately end up with optimal packing (a sort of real-world application of simulated annealing). It’d take a very long time, though.
The results of a small scale experiment with coin packing
The coins in question were approximately 1800 Norwegian 10-øre coins. These have a diameter of 15 mm and a nominal weight of 1.5g, but a single 96 coin layer was weighed on ordinary kitchen scales to weigh 1.50*10[sup]2[/sup]g. It is assumed that various accretions from circulation account for the increased weight.
The coins were put into a 15x14.5 mm container. (The only firm and exactly square container at hand). An presumed optimal packing of 96 coins was found somewhat by trial and error.
2765 grams of coins were poured into the container and shaken, mainly to get an even top surface to measure. Even with a lot of shaking it was hard to get it perfectly even. Measurement at to semi-random points showed a depth of coins of 31mm and 33mm.
The experiment was performed again, but this time the pouring was stopped several times and the container was shaken each time. This time measurement at the same two points showed a depth of precisely 26 mm.
Assuming the weight of a 96 coin layer to be 150 grams the equivalent height with the coins packed as stacks would be 18.433 layers * 1.2mm = 22.12mm. The height of 1.2mm was determined by measuring the height of a single 20 coin stack to 24 mm.
This indicates that the optimal packing of coins is layers with perfect 2D packing. Approximating this by shaking along the way resulted in an increased volume of 17%, while a single pour and extensive post pour shaking resulted in a volume increase of about 44%.
Since the 2D packing left a significant gap along one edge it can be presumed that edge effects worked against rather than for the stacked 2D-packings, and thus this experiment is strong evidence against pouring and shaking giving a non-significant decrease in packing efficiency.
To be clear, the coins are 15 mm in diameter, and the box is only 15 mm on a side? As in, there’s only room for a single stack of coins? Or were the box dimensions meant to be cm?
Yes, centimeters!:smack: I’ve lost the habit of proofreading.
Somewhat anecdotal, but it seems to me that the coins set into rolls are more dense than loose coins. A friend of mine owns an apartment complex with a few coin-operated washers and dryers. Once a month the coins are collected, wrapped, and taken to the bank.
That is, I made the mistake once of showing that I was pretty good at counting and wrapping the coins. So guess who actually gets to do it nowadays. Not all that much fun to do, but no great effort either. It’s something I’d do for a friend, especially since I get occasional free meals out of it <grin>.
Back to the point: when loose, the coins can fill up, say, three fourths of a small sand pail. When wrapped and semi-stacked in the pail, they take up less room, maybe half the pail. The weight of the coins is the same, of course, but it sure feels like those coins are heavier when I carry them into the bank.
I think that when loose, many of the coins are tilted against each other. There’s a lot of the face of each coin that is not laying completely against the face of its neighbor, as it is when wrapped.
Sorry, no objective measurements available.
Stacked , OR perfectly layered, has got to be the most dense packing if the coins have to be horizontal. But you could get more in , if you start with stacked, but put coins slipped in vertically into the gaps left over.
Layered would mean putting the coins in sheets, but the coins in one sheet don’t position exactly above the sheet below. Layered prevents the vertical coins between the horizontal ones… to some extent. There may be variations of layering where the same hole is left , but thats so nearly the same as stacking… stacking could be defined as “layers with coins directly above the coins below” , or loosened to be “well defined layering with dedicating spaces for vertical coins”.
Its got to be that poor layering will create empty voids between two coins of different layers, or coins at angles creating voids in the angles… so unlayered is terrible, and shaking won’t every turn unlayered into layered, and definitely won’t create layered with dedicated verticles filled with vertical coin
Are you talking gaps left over at the edges like in my container? Because stacks will pack close enough to not leave any gaps.
You can also get a greater efficiency with a mixture of different coin sizes than you can with uniform coins.
Beautiful. Thank you.
The result of this secondary experiment clearly shows that this newfangled SI thingy isn’t *quite *as foolproof a units system as the most rabid proponents would have it. Long live furlongs and firkins!! And irrational conversion factors! 
On a serious note, I’d expect that for round coins in a square box only 10x their diameter the edge effects are huge. These sizes are decent proxies for most people’s real world experience with coins and containers, so the experiment is well chosen and highly relevant to them.
But would not be very close to the mathematical limits that would apply to industrial scale stuff like a machine that makes 3/8" flat washers by the ton and dumps them into a receiving bin 10 feet on a side and 20 feet tall.
This effect would be in addition to the point in Chronos’ latest post that varying coin sizes will improve the performance of random packing more than it will deliberately designed optimum packing.