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I have two large identical chests of exactly the same shape, volume, etc. One is completely full of nickels (each worth 5 cents), and the other is exactly half full of dimes (each worth 10 cents). Which chest is worth more money?

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    $\begingroup$ If the chest is half full, I believe dimes are smaller than the 5c coins in the US right? So if the dime were equal size to the N (worst case) - same worth, so if you can pack 1 more dime in there.... So by volume: the half-chest of dimes surely? What do we know about packing? $\endgroup$ – Alec Teal May 28 '15 at 14:18
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    $\begingroup$ Even if nickles and dimes were of equal weight and dimensions, and the monetary value in both chests was the same, I would rather drag home half a chest than a full chest. $\endgroup$ – Dennis Jaheruddin May 29 '15 at 9:34
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    $\begingroup$ It doesn't establish the last part of the necessary information - what is in the other half of the second chest? $\endgroup$ – Glen O May 29 '15 at 12:18
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    $\begingroup$ @GlenO Stacks upon stacks of logic puzzles. $\endgroup$ – Zibbobz May 29 '15 at 13:30
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    $\begingroup$ Packing is also important. You could fill two chests completely (meaning you can't add any more) with nickels and have wildly different numbers of coins in each chest, if you fill them in different ways. $\endgroup$ – Brian Gordon May 30 '15 at 0:11
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This required to get my wallet but it gave me the answer:

The chest half full of dimes is worth more, this because a dime is smaller than a nickle, so it would occupy less space and there would be more dimes in a full chest than there would be in a full chest of nickels. Conclusion: pick the dimes, It's worth more

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    $\begingroup$ The chest of dimes would be worth at least twice as much since dimes are both smaller in diameter and about half the thickness. More math would need to be done to verify exactly how much difference there will be. $\endgroup$ – Jason Hutchinson May 28 '15 at 19:48
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Here's another way to see it:

The chests themselves are worth exactly the same, as they are identical. The contents, on the other hand, might differ.

EDIT: As some people have pointed out, this is an even larger stretch than before if one considers the title as well as the question itself. I didn't.

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    $\begingroup$ It doesn't get any more lateral than this! +1 $\endgroup$ – dmg May 28 '15 at 12:51
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    $\begingroup$ @dmg lateral by way of literal $\endgroup$ – JLee May 28 '15 at 14:23
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    $\begingroup$ That's cute, and got a smile from me, but it's a stretch. In regular language it is clearly understood that the value of a container includes its contents, unless otherwise stated. (Is 50 cents a good price for a bag of chips? Only if it includes buying the chips inside it :) $\endgroup$ – Starman May 28 '15 at 14:39
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    $\begingroup$ The title is confusing. The chest itself would be worth less if it were half a chest like the title says, but this answer is good with regard to the question (not the title). $\endgroup$ – ps2goat May 28 '15 at 14:57
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    $\begingroup$ It also says the chests are identical. If half a chest is identical to a whole chest, it must be a degenerate zero-volume chest, so neither one has any contents to consider. $\endgroup$ – Anon May 28 '15 at 15:17
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It's not that easy!

Dimes have a nominal value of 10 cents, whereas the material value is only about 5.6 cents. Which means the nominal value is decisive.

Nickels have a nominal value of 5 cents but a material value of 10.09 cents (source: Wikipedia). Which means that the material value is decisive (makes you wonder why people aren't buying large amounts of nickels and reselling the metal...).

Further, dimes are not only smaller in diameter than nickels but also slimmer (17.9mm versus 21.2mm diameter and 1.35 versus 1.95mm). That means that the same volume can hold a larger number of dimes than it could hold nickels. It is not possible to tell exactly how many more nickels will fit into the same volume (or into half that volume, for that matter) since this greatly depends not only on stacking, but also on implementation details of the surrounding box. If the box is just half a millimeter too small to exactly fit a complete row of coins, you're drifting far away from the optimal packing, for example.

However, in order to answer the question, it is not necessary at all to know the exact number of coins that will fit. Since both coins are (almost exactly) worth the same, you must have the same number of coins in either case. That is, you must be able to fit the same number of dimes into half the volume, or twice the number of dimes into the same volume. This is certainly not the case, you can only fit roughly 1/3 more dimes into the same volume, give or take a few.

Thus, the box of nickels is more valuable.

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    $\begingroup$ I believe that the OP mentioned in the question that nickels are worth five cents and dimes are worth ten cents. However, your answer is pretty interesting if you ignore the conditions stated in the question. $\endgroup$ – mmking May 28 '15 at 15:49
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    $\begingroup$ @mmking: Well, there's the tag "lateral thinking", so I thought melting the nickles and selling the metal (disregarding nominal value) would be a valid approach :-) $\endgroup$ – Damon May 28 '15 at 15:55
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    $\begingroup$ makes you wonder why people aren't buying large amounts of nickels and resell the metal... because it is illegal i think. $\endgroup$ – Cthulhu May 28 '15 at 17:27
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    $\begingroup$ Section 331 of Title 18 of the United States code provides criminal penalties for anyone who fraudulently alters, defaces, mutilates impairs, diminishes, falsifies, scales, or lightens any of the coins coined at the Mints of the United States. This statute means that you may be violating the law if you change the appearance of the coin and fraudulently represent it to be other than the altered coin that it is. As a matter of policy, the Mint does not promote coloring, plating or altering U.S. coinage: however, there are no sanctions against such activity absent fraudulent intent. $\endgroup$ – Jason Hutchinson May 28 '15 at 19:52
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    $\begingroup$ @JasonHutchinson It's not a "legal gray area" by far -- melting down U.S. cents and nickels is punishable by $10k fine and up to 5 years in jail. usmint.gov/pressroom/?action=press_release&ID=771 $\endgroup$ – Atsby May 29 '15 at 1:47
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Nickel value per volume:

$$\frac{5 ¢}{\pi(\frac{21.21}{2}\ \text{mm})^2 * 1.95\ \text{mm}} = \frac{7.257 ¢}{\text{mL}}$$

Dime:

$$\frac{10 ¢}{\pi(\frac{17.91}{2}\ \text{mm})^2 * 1.35\ \text{mm}} = \frac{29.40 ¢}{\text{mL}}$$

So, for large volumes with reasonable dimensions$\dagger$, if you had twice the volume of nickels as dimes, the nickels would worth about half as much as the dimes.

$\dagger$ I say reasonable dimensions because fringe effects come into play if you have to pay attention to the coins bumping into the wall.

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    $\begingroup$ You haven't taken into account packing density in your calculation. It will have the same net effect on both values, assuming an arbitrarily large volume compared to the objects being packed, scaling by a ratio of ~0.9069. (Exact value: pi/(2*root(3))) $\endgroup$ – Andrew Coonce May 28 '15 at 22:18
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    $\begingroup$ Also, I think you meant the dime equation to be $10¢/V$, not $5¢/V$. :-) $\endgroup$ – Andrew Coonce May 28 '15 at 22:28
  • $\begingroup$ @AndrewCoonce, as you say, packing density affects both values by the same fraction, so doesn't change which is more valuable. $\endgroup$ – user1717828 Dec 6 '15 at 17:35
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It is heavily dependent on the rarity and collectibility of the coins involved. As we are talking about worth, not value, it is subject to much more than just the dollar amount put on the two types of coins. As the problem statement gives no real indicators of worth, I'm going to call BS and say that there ins't enough information.

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    $\begingroup$ The OP very clearly states the worth of each coin. $\endgroup$ – Nuclear Wang Jun 1 '15 at 21:32

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