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Many people are familiar with the story of Archimedes submerging a crown in water to identify if the volume of the crown was consistent with the volume of the same amount of gold. The test was to see if a lighter cheaper metal (like silver) was included in crown. If the crown maker substitutes silver for some of the gold, they make a sinister profit at the risk of their life.

Platinum is heavier than gold, silver is lighter. Using the values as of January of 2013, could any combination of metals be used to create a crown that would have the same volume as a solid gold crown, with a cost effective benefit for the crown maker?

Using the values of metals in 287 BCE – 212 BCE, could it be cost effective for the crown maker.

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    $\begingroup$ This question appears to be off-topic because it is about metallurgy, not about puzzles $\endgroup$ – durron597 May 19 '14 at 16:58
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    $\begingroup$ @durron597 How is it different than Twelve balls and a scale? $\endgroup$ – James Jenkins May 19 '14 at 18:31
  • $\begingroup$ You're asking about the metallurgical properties of various metals. I would expect this question on something like chemistry stack exchange. The other problem can be solved with logic, not outside knowledge. $\endgroup$ – durron597 May 19 '14 at 18:34
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    $\begingroup$ @durron597 it is a weight/cost problem. See the answer by SF below $\endgroup$ – James Jenkins May 19 '14 at 18:36
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    $\begingroup$ @durron597 as for outside knowledge how is it different than End of the world - on a Sunday? $\endgroup$ – James Jenkins May 19 '14 at 19:10
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Nowadays, depleted uranium, with around \$5/gram and 19.1 g/cm3 density could be used to replace some of the gold (probably as insert/skeleton of the crown, as both hardness and appearance wouldn't fool anyone) with gold at \$45/gram at 19.3 g/cm^3 - especially that pure gold would be very rare and it would be hard to discover such minimal difference in density, and adding a little platinum could balance it out to exact density (especially that the price is only marginally higher than gold, so 1g platinum + 1g DU costs way less than 2g of gold, and weighs more).

But then, back in the past, uranium was nearly unobtainable, and there was definitely no technology of purifying it sufficiently (nor producing depleted... using normal uranium as from uranium ore to make a crown would be one really sinister method of murder...)

Now then, back in the past, platinum was called "young silver", sometimes mined alongside real silver. It was deemed worthless though, and buried in the ground "to lie another couple hundred years till it matures into real silver". Therefore, while the 2013 values of using platinum + silver (for example) instead of gold wouldn't earn you any considerable benefit, taking the prices from Archimedes' times, this would be quite the method.

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    $\begingroup$ I'm pretty sure that 1g of platinum + 1g of DU weighs the same as 2g of gold. $\endgroup$ – Peter Taylor Jul 3 '14 at 15:35

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