you can't just check density like that, come on. platinum (which apparently trades around the same price as gold) - has a density of 21.45 [1] which more than compensates. If you want 100 grams of a tungsten-platinum mix to match the weight/volume of 100 grams of gold, this is how many grams you get to use of tungsten:
19.3 * 100 = 19.25 * x + 21.4(100-x)
gold is left, mix is right (a sum of the tungsten and platinum parts). I solve for 100 grams because voila, it's now a percentage. That solves to x = 97.7273 (you get to use 97.7273 grams of tungsten) or you just need 2.2727% platinum. (which recall costs about the same as gold.) so if this were a great test people could just use an alloy, which apparently exist:
19.3 * 100 = 19.25 * x + 21.4(100-x)
gold is left, mix is right (a sum of the tungsten and platinum parts). I solve for 100 grams because voila, it's now a percentage. That solves to x = 97.7273 (you get to use 97.7273 grams of tungsten) or you just need 2.2727% platinum. (which recall costs about the same as gold.) so if this were a great test people could just use an alloy, which apparently exist:
https://www.google.com/search?q=tungsten+platinum+alloy
and target the exact same density as gold. tungsten alone seems close enough for them, at least, according to the author.
[1] http://en.wikipedia.org/wiki/Platinum