# Urns with marbles and gems

In an urn there are three spherical marbles, a red, a green and a blue. Only two of them contain a gemstone inside, while the third is empty. We are not allowed to see or touch them, but we are given the following information about them: The smaller in size between the red and green is the heavier of the two that contain the gem. The lighter between the green and blue is the smaller in size of the two that contain the gem. The larger in size between the red and blue is the lighter of the two that contain the gem. Find the ones that contain the gems. Which marble is the larger and which is the lighter?

I have tried the following: Let's name them $$R$$, $$G$$ and $$B$$. Let's use the symbol "$$\ll$$" for the smaller in size and "$$<$$" for the lighter.

Then we have:
$$R \ll G, R > G,$$
$$G < B, G \ll B,$$
$$R \gg B, R < B$$

So we have $$B \ll R \ll G$$ and $$G < R < B$$ but isn't this contradictory with the second fact?

Apparently I am missing something...

• You don't know that $R<<G$. You only know that if $R<<G$ then $R>G$. If $G<<R$ then $G>R$. Similarly for the other statements. – Derek Elkins Dec 3 '18 at 21:04
• @DerekElkins: that is not quite correct. if $R \lt \lt G$ then $R$ has a gem and $R \gt$ the other that has a gem. We don't know from the first fact that the gems are in red and green. – Ross Millikan Dec 3 '18 at 21:10
• I made an edit in the last line. – Eduardo Juan Ramirez Dec 3 '18 at 21:25
• @RossMillikan Oops. You're right. – Derek Elkins Dec 3 '18 at 21:41