Two logic problems - Math Central

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Question from Grace:

1. You are an archaeologist that has just unearthed a long-sought triplet
of ancient treasure chests. One chest is plated with silver, one with gold,
and one with bronze. According to legend, one of the three chests is filled with
great treasure, whereas the other two chests house a man-eating python that
can rip your head off. Faced with a dilemma, you then notice that
there are inscriptions on the chests:

Silver Chest - Treasure is in this Chest.
Gold Chest - Treasure is not in this Chest.
Bronze Chest - Treasure is not in the Gold Chest

.
You know that at least one of the inscriptions is true,
and at least one of the inscriptions is false. Which chest do you open?

(a) Silver (b) Gold (c) Bronze

2. There are 3 black hats and 2 white hats in a box. Three men
(we will call them A, B, & C) each reach into the box and place one of the hats
on his own head. They cannot see what color hat they have chosen.
The men are situated in a way that A can see the hats on B & C's heads,
B can only see the hat on C's head and C cannot see any hats.
When A is asked if he knows the color of the hat he is wearing, he says no.
When B is asked if he knows the color of the hat he is wearing he says no.
When C is asked if he knows the color of the hat he is wearing he says yes
and he is correct. What color hat is C wearing and how can this be?

(a) Black (b) White

Hi Grace,

I can get you started on these problems.

For the first problem suppose that the treasure is in the silver chest.

Is the first statement true? Yes the treasure is in the silver chest.
Is the second statement true? Yes because the treasure is not in the gold chest.
Is the third statement true? Yes because the treasure is not in the gold chest.

Hence if the treasure is in the silver chest all three statements are true. But you know that one of the statements is false so the treasure is not in the silver chest.

Now suppose the treasure is in the gold chest. Which of the statements are true and which are false.

For the second problem A says he can not tell what colour hat he is wearing. He can see what colour hats B and C are wearing. There are only two white hats so if B and C were wearing white hats then A would know his was black. Thus either B or C or perhaps both is wearing a black hat.

Have fun with this,
Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.