# Logic puzzles

Logic puzzles are used as basic exercises not only in mathematics but in law. The Law School Admission Test (LSAT) is full of logical puzzles, and the preparatory materials are called Logic Games (see, e.g., sample logic game or more logic games). Such puzzles often appear in recreational mathematics magazines and books. The following are drawn from that tradition.

## Introducing logic puzzles

Outside of school, puzzles don't usually come with instruction; we see the puzzle and make a choice either to ignore it or to puzzle it out on our own. In class, there's less choice, so some introduction is valuable for students who might not be intrepid enough to tackle a puzzle. Because the purpose of the puzzles is to strengthen the Let's-see-if-this-works approach to problem solving, a good strategy to start with is "Guess an answer, and test it to see if it works."

The following made-up dialogue about the first puzzle in the list below illustrates two things:

• the teaching of some important academic language: the language of if...then; and
• a common trap that beginners, especially beginners who don't feel confident about their thinking, fall into.
Start by telling the students about the Liar and Truthteller families. It is important to know that we "normal people" mostly tell the truth and sometimes lie, so we aren't totally reliable. But these special families, the Liars and Truthtellers always lie or tell the truth, depending on their family. They are 100% reliable!

If students have no idea, suggest that they just guess about the speaker. There are two possibilities: Adam's a Liar, or he's a Truthteller.

• Student: I think Adam is a Liar.
• Teacher: Let's check it out. If Adam is lying, then... What?
• Student: They're not both Liars.
• Teacher: Right. If Adam is lying, then they're not both Liars. And what's Adam?
• Student: Well, if he lied, then he has to be a Liar.
• Teacher: And do we know anything about Alec?
• Student: Well, they can't both be Liars, so he must be a Truthteller.

Alternatively:

• Student: I think Adam is a Truthteller.
• Teacher: Let's check it out. If Adam is telling the truth, then... What?
• Student: Wait! If he's telling the truth, then he can't be a Liar!
• Another student: But he's just said they are both Liars.

People -- adults and students -- who are new to these puzzles often get stumped at this point. Especially if they are not confident about their reasoning, they may attribute the contradiction to bad reasoning. Rather than recognizing that their logic -- "If Adam is a Truthteller, then he can't say he's a Liar" -- is absolutely correct, which solves the puzzle, they feel confused. The then part of their logic is correct, but they stopped a moment too soon. At this point, students may undo their own good logic unless it is validated.

• Teacher: Right! Perfect! If Adam is telling the truth, then he can't say he's a Liar, but he did say he's a Liar. So?

The teacher resists giving the next conclusion, but does validate the thinking so far.

• Student: So he can't be telling the truth! He's a Liar!
Students are rarely so articulate at the very start, but sometimes say things like "He says they're both Liars. If they are, then he's telling the truth, so I think he's a Truthteller," which seems utterly illogical, but that's what makes these puzzles puzzles. It's very easy to get caught in contradicting oneself until one is used to sorting out the logic.

## Liar and Truth-teller puzzles

People in the Liar family always tell lies. Never, not even by accident, do they ever tell the truth. Never! The Truthteller family is just as trustworthy. You can depend on them to tell the truth, the whole truth, and nothing but the truth, always. The tough thing is that there is no way to tell these families apart just by looking at them. So, you have to be logical.
2. Then you meet Alex and Ali. Alex says “Exactly one of us is a Liar.” What family is Ali from?
3. Now you meet Antoine and Ashley. Ashley says “We are both from the same family.” What can you figure out about Antoine?
4. Ariel and Ben come by. Ariel says “Exactly one of us is a Truthteller.” What family is Ben from? Can you tell what family Ariel is from?
5. Chris and Claire show up, and Chris says “At least one of us is a Truthteller.” What, if anything, can you say about their families?
6. Suppose Claire also speaks, and says “Chris is telling the truth!” Then what can you say about their families?
7. David and Devon pass by. David says “At least one of us is a Liar.” What, if anything, can you say about their families?
8. Elizabeth and Ellie are next. Elizabeth makes exactly one statement, which might be true, or it might be a lie. She says: “I am a Liar and Ellie is a Truthteller.” What, if anything, can you figure out about Elizabeth and Ellie? (This is tough!)
9. Gregory and Hannah show up, and Hannah makes one complicated statement. She says “It's not true that Gregory and I are both Truthtellers!” What families are they from?
10. Izzie says “I'm a Truthteller.” Jessica says “We're both from the same family.” What do you think?
11. John and Jennifer happen by. John says “We are from different families,” and Jennifer says “John's lying!” Is either of them telling the truth?
12. You meet Josh and Karina in the garden. Josh says “We are both Truthtellers,” and Karina says “Josh isn't lying.” What can you figure out about their families?

Harder puzzles:

1. Kaitlyn and Lorna walk up to you. Kaitlyn says “I always tell the truth.” Lorna says “Kaitlyn is lying right now!” What can you work out about these two?
2. Melissa and Michael are both doing their homework when you arrive. Melissa says “We are from different families.” Michael says “We are from the same families, and Melissa is lying!” What are their real families?
3. Nathalie and Nick are both doing their homework when you arrive. Nathalie says “We are from different families.” Nick says “We are from the same family, and I am lying.” What are their real families?
4. Nicole and Sarah make these statements. Nicole: “We are not both Truthtellers.” Sarah: “She's lying!” What do you think?
5. Serena and Spencer show up. Serena says “Exactly one of us is telling the truth,” and Spencer says “Exactly one of us is lying.” What do you think?
6. Steven, Tori, and Zack all show up together, all talking at once. Steven says “Tori always lies.” Tori says “Zack is quite truthful.” Zack says, rather enigmatically, “Either I am from the Liar family or I am from the Truthteller family.” Which families are the three from?

## Variations on the Liar and Truth-teller puzzles

The following puzzle is adapted from Raymond Smullyan's The Lady and the Tiger.[2] It doesn't look like a Liar and Truthteller puzzle, but it is really the same. In this case, the question is "Which sign is telling the truth?"

The king of a far away land decided on the perfect way to try his prisoners. The prisoner would have to choose between two rooms, one of which contains a great banquet and the other of which contains a tiger. If he chooses the former, he gets to dine at the banquet, and is let free; if he chooses the latter, the tiger gets to dine on him! The king designed this ordeal as a test. The king put signs on the doors of the rooms, but the signs posed a puzzle. If the prisoner reasoned logically, he could figure out which room to choose, saving his life, and giving him a great banquet, too!
If you were the prisoner, which door would you open (assuming, of course, that you prefer eating to being eaten)?
1. The king brought the first prisoner to the two doors and said said “One of these signs is true, but the other is false. Choose well.”
Door 1: This room has the banquet, and the other room has the tiger.
Door 2: One of these two rooms has the banquet, and the other room has the tiger.
2. (more puzzles to come)

## References

1. Case, Adam. Who Tells the Truth?. Stradbroke, Norfolk, England: Tarquin. 1990.
2. The Lady and the Tiger. Smullyan, R.