SAT1000 - P935

From the statement, "Alice looked at Betty’s chest and said: "My chest and hers don't both contain a silver coin,"" we can conclude that Cathy does not have the chest with the gold coin and rock. Cathy also says that she does not have the chest with the silver coin and rock. Therefore, Cathy must have the chest with the gold coin and silver coin.

From the statement, "Betty looked at Cathy’s chest and said: "My chest and hers don't both contain a gold coin,"" and knowing that Cathy has a chest with a gold coin, we can conclude that Betty must have the chest with the silver coin and rock.

That leaves Alice with the chest with the gold coin and rock .

A = Alice. B = Betty. C = Cathy. S = Silver. G = Gold. R = Rock.

From the third statement, Cathy either has S or R, but not both. From the given three treasure chests, only two fit this condition: G and S, or G and R. Regardless of which, G is necessary.

From the second statement, both B & C cannot have G. Thus, B could not possibly have G. Naturally, since there are two Gs in total, the 2nd G must belong to A.

Since B has G crossed, the only remaining options left are S and R.

From the first statement, A and B cannot have silver coins. Since B already has one, A cannot possibly have one.

We can fill in the rest from there.

You can check this with the initial conditions just to double confirm. In conclusion, Alice (A) has a gold coin and a rock.

I like working on problems backwards, so let's start.

From the statement, "Cathy said: "What in my chest aren't a silver coin and a rock."", we'll assume that Cathy has a gold coin with a silver coin out of the two options for her, because she doesn't have the a silver coin and a rock. If this assumption is wrong, we can always go back and change it.

Betty said that her and Cathey's chest don't both contain a gold coin. This is important because we know for a fact that Cathy's chest has a gold coin, thus Betty must have the silver coin and a rock.

Know that we know Betty's chest contains a silver coin, we can deduct that Alice's chest must be the one with a gold coin and a rock, because of the statement Alice looked at Betty's chest and said, "My chest and hers don't both contain a silver coin.", and we know Betty's contains a silver coin, thus Alice's must be the chest with a gold coin and a rock .

We have three treasure chests of which 2 contain a gold coin, 2 contain a silver coin and 2 contain a rock.

1. Cathy says she does not have the treasure containing the silver coin and the rock. We still don't know which of the other two treasure chests she has, but we do know one thing for sure: she has to have a gold coin in her chest.
2. From Betty's statement we know only one of the two girls has gold in their chest. We already know Cathy has a gold coin, meaning Betty has to have the chest containing the silver coin and the rock, since it's the only chest that does not contain gold. Now we also know that Alice has to have a gold coin in her chest as well.
3. From Alice's statement we know only one of the two girls has silver in their chest. We already know Betty has a silver coin in her chest, meaning Alice has the chest containing the gold coin and the rock.

From Cathy's statement, we know that one of the other two must have taken the silver-rock chest.

From Betty's statement, since one of the gold combos belongs to either one of Betty or Cathy, then Alice must have had that other gold chest, so the silver-rock combo must have been Betty's.

By Alice own admission, hers doesn't contain any silver coins (since Betty already has them), so the answer must be gold-rock chest.

Statement #1 => either Alice or Betty will have Gold-Rock combo.

Statement #2 => either Betty or Cathy will have Silver-Rock Combo.

Statement #3 => Cathy don't have Silver-Rock Combo

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From #3 & #2 we get "Betty has Silver Rock Combo" (Statement #4)

From #4 & #1 we get "Alice has Gold-Rock Combo"

Tadaaa

• Alice: "My chest and hers (Betty's) don't both contain a silver coin." This means that either Alice or Betty had the chest with $\red{\text{a gold coin and a rock}}$ .
• Betty: "My chest and hers (Cathy's) don't both contain a gold coin." This means that either Betty or Cathy had the chest with $\blue{\text{a silver coin and a rock}}$ .
• Since Cathy: "What in my chest aren't $\blue{\text{a silver coin and a rock}}$ ". This means that Betty had the chest with $\blue{\text{a silver coin and a rock}}$ and Alice was the one having the chest with $\boxed{\red{\text{a gold coin and a rock}}}$ .