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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...

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  • $\begingroup$ 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. $\endgroup$ – Derek Elkins Dec 3 '18 at 21:04
  • $\begingroup$ @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. $\endgroup$ – Ross Millikan Dec 3 '18 at 21:10
  • $\begingroup$ I made an edit in the last line. $\endgroup$ – Eduardo Juan Ramirez Dec 3 '18 at 21:25
  • $\begingroup$ @RossMillikan Oops. You're right. $\endgroup$ – Derek Elkins Dec 3 '18 at 21:41
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You are not transcribing the facts correctly. You have taken the first statement to say that red is smaller than green and red is heavier than green. What it says is that whichever is smaller is the heavier of the two marbles that contain a gem. You therefore cannot use the transitivity to derive the contradiction you did.

I think I would just list the six possibilities for size ranking and see where each leads.

Say they are RGB from large to small. Then green is the heavier of the two that have gems (fact 1), red is the lighter of the two that have gems (fact 3), and blue is empty. Fact 2 now says that as green is the smaller that has a gem it is lighter than blue. Then they are BGR in order of weight. On the assumption that the puzzle is correctly posed, we could stop here and report the answer. Red and green have gems, red is larger and lighter.

To show how a problem can arise, say they are GBR from large to small. Now red is the heavier one with a gem and blue is the lighter one with a gem. Fact 2 says that the smaller one with a gem is blue or green, so we have a contradiction and this order of sizes is not possible.

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