Question9. Suppose you are visiting an island with knights who always tell the truth, knaves who always lie, and jokers who can do either.
You meet three islanders named Ellis, Farin, and Gobi. They all know what the others are (a knight, knave, or joker) and make the following statements:
If exactly one of them is a joker, how many of them are knights?
ELLIS
FARIN
Ellis is a joker! 9) $\qquad$ $\square$
10. There are 3 boxes, exactly one of which contains gold. You can keep the gold if you pick the correct box! On each box there is a statement, exactly one of which is true.
The gold is in this box
1
The gold is not in this box
2
The gold is not in box 1
3
Which box has the gold? 10) $\qquad$

STEP 1

1. Knights always tell the truth.
2. Knaves always lie.
3. Jokers can either lie or tell the truth.
4. Exactly one of the three islanders is a joker.
5. Exactly one of the statements on the boxes is true.
_HIGH_LEVEL_APPROACH_ for Problem 9:
1. Analyze the statements made by Ellis, Farin, and Gobi.
2. Determine the roles of each islander based on the given conditions.
3. Count the number of knights.
_HIGH_LEVEL_APPROACH_ for Problem 10:
1. Analyze the statements on each box.
2. Determine which statement is true.
3. Identify the box containing the gold.

STEP 2

STEP 3

(Problem 9) Analyze Ellis's statement: "Ellis is a joker!"
- If Ellis is a knight, then the statement is false, which is a contradiction. - If Ellis is a knave, then the statement is true, which is a contradiction. - Therefore, Ellis must be the joker.

STEP 4

(Problem 9) Analyze Farin's statement: "Farin is a knight!"
- If Farin is a knight, the statement is true. - If Farin is a knave, the statement is false. - Since Ellis is the joker, Farin must be a knight.

STEP 5

(Problem 9) Since Ellis is the joker and Farin is a knight, Gobi must be a knave.
Count the number of knights: There is 1 knight (Farin).
The number of knights is:
$$\boxed{1}$$

STEP 6

(Problem 10) Analyze the statements on each box:
- Box 1: "The gold is in this box." - Box 2: "The gold is not in this box." - Box 3: "The gold is not in box 1."

STEP 7

(Problem 10) Determine which statement is true:
- If Box 1's statement is true, then Box 2 and Box 3's statements must be false, which is consistent. - If Box 2's statement is true, then Box 1 and Box 3's statements must be false, which is a contradiction. - If Box 3's statement is true, then Box 1 and Box 2's statements must be false, which is a contradiction.

STEP 8

(Problem 10) Since Box 1's statement is the only one that can be true, the gold is in Box 1.
The box with the gold is:
$$\boxed{1}$$