Logic • Hard 1 Set

Constraint puzzle: A code uses 3 colors. You know:
1) green is used.
2) If blue is used, then red is not used.
3) yellow is not used.
Which set of colors is possible?

We must include green and exclude yellow.

If we include blue, we must exclude red.

The only option that satisfies all constraints is: green, blue, purple.

Three people Zoe, Mia, and Ivy each make one statement. Exactly one of them is true.

Zoe: “Mia is lying.”
Mia: “Ivy is lying.”
Ivy: “Zoe is lying.”
Who is telling the truth?

Try each possibility:

1. If Zoe is true, then Mia is lying. That forces Mia's statement to be false, which forces Ivy's statement to be true — contradiction.
2. If Mia is true, then Ivy is lying. That forces Ivy's statement to be false, which forces Zoe's statement to be true — contradiction.
3. If Ivy is true, then Zoe is lying, which forces Zoe's statement to be false, which forces Mia's statement to be true — contradiction.

No assignment makes exactly one statement true, so the correct answer is Impossible to determine (the setup is inconsistent).

Three people Ivy, Mia, and Ava each make one statement. Exactly one of them is true.

Ivy: “Mia is lying.”
Mia: “Ava is lying.”
Ava: “Ivy is lying.”
Who is telling the truth?

Try each possibility:

1. If Ivy is true, then Mia is lying. That forces Mia's statement to be false, which forces Ava's statement to be true — contradiction.
2. If Mia is true, then Ava is lying. That forces Ava's statement to be false, which forces Ivy's statement to be true — contradiction.
3. If Ava is true, then Ivy is lying, which forces Ivy's statement to be false, which forces Mia's statement to be true — contradiction.

No assignment makes exactly one statement true, so the correct answer is Impossible to determine (the setup is inconsistent).

Constraint puzzle: A code uses 3 colors. You know:
1) blue is used.
2) If green is used, then purple is not used.
3) red is not used.
Which set of colors is possible?

We must include blue and exclude red.

If we include green, we must exclude purple.

The only option that satisfies all constraints is: blue, green, yellow.

Constraint puzzle: A code uses 3 colors. You know:
1) red is used.
2) If green is used, then blue is not used.
3) yellow is not used.
Which set of colors is possible?

We must include red and exclude yellow.

If we include green, we must exclude blue.

The only option that satisfies all constraints is: red, green, purple.

Constraint puzzle: A code uses 3 colors. You know:
1) purple is used.
2) If blue is used, then green is not used.
3) red is not used.
Which set of colors is possible?

We must include purple and exclude red.

If we include blue, we must exclude green.

The only option that satisfies all constraints is: purple, blue, yellow.

Constraint puzzle: A code uses 3 colors. You know:
1) red is used.
2) If green is used, then purple is not used.
3) blue is not used.
Which set of colors is possible?

We must include red and exclude blue.

If we include green, we must exclude purple.

The only option that satisfies all constraints is: red, green, yellow.

Three people Ethan, Ava, and Liam each make one statement. Exactly one of them is true.

Ethan: “Ava is lying.”
Ava: “Liam is lying.”
Liam: “Ethan is lying.”
Who is telling the truth?

Try each possibility:

1. If Ethan is true, then Ava is lying. That forces Ava's statement to be false, which forces Liam's statement to be true — contradiction.
2. If Ava is true, then Liam is lying. That forces Liam's statement to be false, which forces Ethan's statement to be true — contradiction.
3. If Liam is true, then Ethan is lying, which forces Ethan's statement to be false, which forces Ava's statement to be true — contradiction.

No assignment makes exactly one statement true, so the correct answer is Impossible to determine (the setup is inconsistent).

Constraint puzzle: A code uses 3 colors. You know:
1) green is used.
2) If yellow is used, then purple is not used.
3) red is not used.
Which set of colors is possible?

We must include green and exclude red.

If we include yellow, we must exclude purple.

The only option that satisfies all constraints is: green, yellow, blue.

Logic check: All pilots are night owls. Some night owls are introverts. Which statement must be true?

Given: All pilots are night owls. Some night owls are introverts.

Key idea: “Some night owls are introverts” means there exists at least one person who is both night owls and introverts — so it is automatically true that some introverts are night owls.

Nothing guarantees that any pilots are introverts, so option 1 is not required.