Logic • Medium 1 Set

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

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

Try each possibility:

1. If Ivy is true, then Ava is lying. That forces Ava's statement to be false, which forces Ethan's statement to be true — contradiction.
2. If Ava is true, then Ethan is lying. That forces Ethan's statement to be false, which forces Ivy's statement to be true — contradiction.
3. If Ethan is true, then Ivy is lying, which forces Ivy'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) yellow is used.
2) If blue is used, then purple is not used.
3) green is not used.
Which set of colors is possible?

We must include yellow and exclude green.

If we include blue, we must exclude purple.

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

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).

Logic check: All runners are problem solvers. Some problem solvers are students. Which statement must be true?

Given: All runners are problem solvers. Some problem solvers are students.

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

Nothing guarantees that any runners are students, so option 1 is not required.

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) 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) 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.

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).

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.