That's a children's parable. You can easily defeat the test by using two metals, one denser than gold and one lighter, whose combination matches the density of gold, and then putting a veneer of gold on top of it.
You can nowadays look at a 3D-plot of internal density in an X-Ray computed tomography. On the chance that somebody mixed up an alloy that has the exact same density and X-Ray density as gold, you can try X-Ray absorption spectra which depend on nuclear resonances of different nuclei, so would conclusively prove presence and amount of non-Gold material.
And you can do activation analysis, where you activate the ingot with neutron radiation such that Gold isotopes form and decay. You then measure the decay gamma radiation spectrum and look for decay lines that are not gold but another material. This is non-destructive, but will make that ingot slightly radioactive for a while, but nothing that a few weeks of patience to wait for the decay can't fix.
Oh, and nuclear magnetic resonance (NMR) spectra would also work, but usually those machines are for very small samples in the milligram region. No idea whether there are some that could fit a gold ingot.
You know slightly different NMR machines are used to scan entire humans, right?
They only really differ in scan pattern for this aspect. That's IIUC just a software change.
Afaik not really. The ones that scan humans are tuned for a specific proton resonance frequency, so they basically measure hydrogen density. But you get a 3D-Plot because they do a computed tomography of that density.
The chemistry NMRs are spectrographs, so you dial through a whole frequency spectrum and look at the reflected/transmitted signal. With that you get different peaks for different nuclei plus some deviations for crystal or molecular structure.
But I have to admit, that I'm a little rusty on that topic, so you might be right anyways.
Yeah right I looked again and also: MRI doesn't really work for highly conductive (bulk) samples: you can't get the RF in/out of the conductor due to skin effect.
That wouldn’t work in Archimedes’ time because gold was the densest metal available then.
It could be done now but most of the candidate elements cost more than gold. Uranium looks feasible if the gold veneer will block the radiation which otherwise might tip off the recipient.
