# Which would you prefer: A half a barrel of dimes or a barrel of nickels?

Is there a way to mathmatically determine the answer to the question: Which would you prefer: A half a barrel of dimes or a barrel of nickels?

It seems to me that there isn’t enough information. But I haven’t had any mathmatics in many years.

Stacking geometry notwithstanding, you can answer this problem on the basis of the relative volume of the two coins. Assuming that the barrel referred to in each case is of the same volume, look a a dime, and then a nickel - you will notice that the volume of a dime is considerably less than half of the volume of a nickel. When you fill a barrel half full of dimes, it stands to reason that the value of these coins will be greater than a full barrel full of nickels.

Dimes and quarters, on the other hand…

I’ll take a half-barrel that is half-full half-way right on up to the half-way point with dimes, but only if half of it’s filled.

Dimes, I think.

Dimes, most obviously - Not only can you get at least twice as many in the same volume, they’re twice as valuable coin-for-coin, meaning that the barrel of dimes would be at least four times more valuable than the nickels.

I had no luck googling for the volume of a dime or nickel, but I since I happened to have my trusty dial calipers sitting on my desk, I was able to measure the diameter and thickness of a “representitive” dime and nickel I had in my change bucket.

The dime measured 0.796" in diameter and 0.050" thick. The nickel measured 0.838" in diameter and 0.075" thick.

Since cylinders don’t pack well, I calculated volume as if the surface was a square rather than a circle. (This is where I may be off, is the packing better if they’re arranged in “hexagons”). The dime’s “volume” is 0.0316 in^3 and the nickel’s “volume” is 0.0527 in^3.

Since the dime’s volume is greater than half the nickel’s volume, but it is only twice the value, I think the better choice is the full barrel of nickels.

Interesting that everyone else has chosen dimes while I’ve been off measuring…

It seems that relative volumes on this scale are difficult to estimate.

Read the OP again. You pick either:

1. A Full barrel of nickels

-or-

1. A **HALF ** barrel of dimes.

I’d still go with the dimes.

Explain that please?? Even if they have the same volume, as long as the dime is twice the value of the nickel the barrels would be of equivalent value. ANY space savings with the dime translates into an advantage.

Of course, that’s only if you’re interested in maximizing cash value. If your personal preferences depend on other criteria, that’s your decision.

Well, if this was in the mix, I’d take the nickels.

This reminds me of a joke I just read:

There’s a little fellow named Junior who hangs out at the local grocery store. The manager doesn’t know what Junior’s problem is, but the boys like to tease him. They say he is two bricks short of a load, or his elevator doesn’t go all the way to the top.

To prove it, sometimes the boys offer Junior his choice between a nickel and a dime. He always takes the nickel, they say, because it’s bigger.

One day after Junior grabbed the nickel, the store manager got him off to one side and said, “Junior, those boys are making fun of you. They think you don’t know the dime is worth more than the nickel. Are you grabbing the nickel because it’s bigger, or what?”

Junior said, “Well, if I took the dime, they’d quit doing it!”

You sure you measured right? The US Mint says the diameter of a dime is about 0.705 in. Moreover, this page claims that there are about 5250 nickels per gallon, and about 10500 dimes, for an exact 2:1 ratio. [And, not on topic, but interesting: dimes, quarters, and half dollars work out to the same amount of money per gallon.]

Exactly. Since a dime is not larger than a nickel, I can’t see any way that a full barrel of nickels would ever have more monetary value than the half-barrel of dimes (assuming we’re talking the same size barrel here).

At worst, you’d have half as many dimes as nickels (same monetary value), and more likely you’d have at least some extra dimes.

And the half-barrel of dimes would be lighter – easier to carry to the bank.

Now, if you’ll offer me a half-barrel of paper money, I’ll forget I ever heard about the dimes.

JTC: Since cylinders don’t pack well, I calculated volume as if the surface was a square rather than a circle. (This is where I may be off, is the packing better if they’re arranged in “hexagons”).

Yes, it is. The hexagonal lattice packing (with each circle surrounded by six circles touching it) is the maximally dense packing of uniformly-sized circles.

(Math Phun Phact: The corresponding statement for three dimensions, that hexagonal lattice packing of uniformly-sized spheres is maximally dense, was conjectured by Johannes Kepler back in 1611 but not actually proved until a few years ago, by Thomas Hales.

Physics Phun Phact: some dude whose name I don’t remember experimentally tested the Kepler conjecture by leaving dried peas in a pot of water overnight and noting that their expansion had squeezed them into little dodecahedrons, suggesting that the space-saving distribution they naturally settled into was the hexagonal lattice packing. This illustrates why the Kepler conjecture always used to be introduced with the words “Many mathematicians believe, and all physicists know, that…”

I always get a chuckle out of thinking of that guy counting the little faces on a soggy pea. :))

Weigh the barrels. Let w[sub]1[/sub] be the weight of the barrel of nickels, and w[sub]2[/sub] be the weight of the barrel of dimes. Let d be the weight of a dime, and n the weigh of a nickel.

The value of the barrel of dimes is w[sub]1[/sub]/10d, and the value of the barrel of nickels is w[sub]2[/sub]/5n. If w[sub]1[/sub]/w[sub]2[/sub] > 2d/n, then the barrel of dimes is worth more. Otherwise, it’s the barrel of nickels.

What exactly is the connection between hexagonal lattices for spheres, and dodecahedrons… I don’t believe that dodecahedrons will pack together in a hexagonal lattice… or any lattice very neatly. I could be wrong.

It’s not mathematics, it’s just simple logic.

Assume that dimes and nickels are exactly the same shape.

Fill half a barrel with dimes, and fill a barrel with nickels. By definition, the two are worth the exact same value- the barrel of nickels has twice as many coins, but each coin is worth half the value of a dime.

But, because nickels are larger than dimes, there must be more dimes in the half-barrel than half the number of nickels in the full barrel. Therefore, the half-barrel of dimes is worth more.

chrisk: * I don’t believe that dodecahedrons will pack together in a hexagonal lattice… or any lattice very neatly. I could be wrong.*

I don’t think you’re wrong about regular dodecahedra, but the ones that hexagonal-lattice-packed spheres expand into are rhombic dodecahedra, which are space-filling: