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Logic Puzzles

Truth-tellers/liars, deduction, self-referential statements, ordering puzzles

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Logic PuzzlesAMC 10 Step by Step

Name: Date: October 28, 2026

18 problems, easiest first

What you need to know

Truth-tellers and liars. A truth-teller says only true things; a liar says only false things. So a speaker is a truth-teller exactly when what they say is true. Example: if Kim says "Lee is a liar," then either Kim is truthful and Lee lies, or Kim lies and Lee is truthful. Either way, they are different types.
"Must be true" means true in every situation that fits the facts given. One counterexample (a situation that fits the facts but breaks the statement) rejects it. Example: "every club member plays chess" does not force "every chess player is in the club": one chess player outside the club breaks it.
"Some" means at least one, and it allows "all." Example: "some of my pets are dogs" is still true if every pet is a dog.
"All $A$ are $B$ " does not mean "all $B$ are $A$ ." Knowing that something is a $B$ tells you nothing about whether it is an $A$ . Example: all squares are rectangles, but not all rectangles are squares.
Nobody can say "I am a liar." A truth-teller saying it would be lying, and a liar saying it would be telling the truth.
"We are both liars" can only come from a liar whose partner is a truth-teller. A truth-teller can't say it, since it would make the speaker a liar. So the speaker lies, the statement is false, and the two are not both liars: the partner tells the truth.

How AMC 10 tests it

Rare, usually in positions 5–15, and solved by casework rather than by a formula.
"Which statements must be true?" with three claims labeled I, II, III. Test tiny made-up worlds, and read "some" literally.
Truth-teller/liar puzzles where people talk about each other. Find one claim that settles a relationship in both cases, then follow where it leads.

Standard approaches

First rewrite each sentence as a precise "if ... then ..." statement.
Find the anchor: a statement that tells you the same thing whether the speaker is a truth-teller or a liar.
Split into cases on one person's type and follow the consequences. A contradiction rules that case out.
For "must be true," build the smallest world that fits the facts and see which statements fail.

Worked example

Ava, Ben, Cal, and Dee are each either a truth-teller or a liar. Ava says, "Ben and Cal are both liars." Ben says, "Exactly one of Ava and Dee is a liar." Cal says, "Dee is a truth-teller." Dee says, "Ava is a liar." How many of the four are truth-tellers?

(A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) $4$

Start with Dee, the anchor. If Dee tells the truth, Ava is a liar. If Dee lies, Ava is a truth-teller. In both cases Ava and Dee are different types, so exactly one of them is a liar.

Ben. That is Ben's claim, so Ben is a truth-teller.

Ava. Ava said Ben and Cal are both liars. Ben is not, so Ava's statement is false and Ava is a liar.

Dee and Cal. Since Ava is a liar, Dee's statement is true, so Dee is a truth-teller. Then Cal's statement about Dee is true, so Cal is a truth-teller too.

So the truth-tellers are Ben, Cal, and Dee. The answer is $\boxed{\textbf{(D)}\ 3}$ .

Pitfalls

Reading "some $A$ are $B$ " as "some but not all," or turning "all $A$ are $B$ " around into "all $B$ are $A$ ."
Stopping at one assignment that works without checking that the other cases fail.
Thinking that when a liar joins two claims with "and," both are false. You only know that at least one is false: a liar who says "I have a cat and a dog" might have a cat.
Using real-world facts that the problem never states.

Traps that recur

Starting from red 1 (which fits anywhere) rather than from the most restricted cards, and getting lost in possibilities. (2003 AMC 10A #24)
Assuming all alternaters answer alike; those who start truthful say no three times, those who start lying say yes three times. (2022 AMC 10A #12)
Assigning Aditi 3 + 4 = 7 before noticing that Oscar's score of 4 already requires cards 1 and 3. (2021 AMC 10B #17)
Assuming Ada must return to seat 5 (the right end) or forgetting that Dee and Edie's swap contributes zero net movement. (2016 AMC 10A #13)

1.2019 AMC 10B #2
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
1. A) $11$
2. B) $15$
3. C) $19$
4. D) $21$
5. E) $27$
2.2017 AMC 10A #6
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which of these statements necessarily follows logically?
1. A) $\text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}$
2. B) $\text{If Lewis did not receive an A, then he got at least one of the multiple choice questions wrong.}$
3. C) $\text{If Lewis got at least one of the multiple choice questions wrong, then he did not receive an A. }$
4. D) $\text{If Lewis received an A, then he got all of the multiple choice questions right.}$
5. E) $\text{If Lewis received an A, then he got at least one of the multiple choice questions right.}$
3.2015 AMC 10B #5
David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$ -person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$ th place. Who finished in $8$ th place?
1. A) $\text{David}$
2. B) $\text{Hikmet}$
3. C) $\text{Jack}$
4. D) $\text{Rand}$
5. E) $\text{Todd}$
4.2007 AMC 10B #5
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
1. A) $\text{All Dramps are Brafs and are Crups.}$
2. B) $\text{All Brafs are Crups and are Dramps.}$
3. C) $\text{All Arogs are Crups and are Dramps.}$
4. D) $\text{All Crups are Arogs and are Brafs.}$
5. E) $\text{All Arogs are Dramps and some Arogs may not be Crups.}$
5.2025 AMC 10A #8
Agnes writes the following four statements on a blank piece of paper.
$\bullet$ At least one of these statements is true.
$\bullet$ At least two of these statements are true.
$\bullet$ At least two of these statements are false.
$\bullet$ At least one of these statements is false.
Each statement is either true or false. How many false statements did Agnes write on the paper?
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
6.2022 AMC 10B #11
All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement "No school bigger than Euclid HS sold more T-shirts than Euclid HS"?
1. A)All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS.
2. B)No school that sold more T-shirts than Euclid HS is bigger than Euclid HS.
3. C)All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS.
4. D)All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS.
5. E)All schools smaller than Euclid HS sold more T-shirts than Euclid HS.
7.2021 AMC 10A #7
Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that
- all of his happy snakes can add,
- none of his purple snakes can subtract, and
- all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
1. A)Purple snakes can add.
2. B)Purple snakes are happy.
3. C)Snakes that can add are purple.
4. D)Happy snakes are not purple.
5. E)Happy snakes can't subtract.
8.2021 AMC Fall 10B #10
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
1. A) $27$
2. B) $37$
3. C) $47$
4. D) $57$
5. E) $67$
9.2015 AMC 10B #6
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
1. A) $\text{Sunday}$
2. B) $\text{Tuesday}$
3. C) $\text{Thursday}$
4. D) $\text{Friday}$
5. E) $\text{Saturday}$
10.2013 AMC 10A #6
Joey and his five brothers are ages 3, 5, 7, 9, 11, and 13. One afternoon two of his brothers whose ages sum to 16 went to the movies, two brothers younger than 10 went to play baseball, and Joey and the 5-year-old stayed home. How old is Joey?
1. A) $3$
2. B) $7$
3. C) $9$
4. D) $11$
5. E) $13$
11.2011 AMC 10B #8
At a certain beach if it is at least $80^{\circ} F$ and sunny, then the beach will be crowded. On June 10 the beach was not crowded. What can be concluded about the weather conditions on June 10?
1. A) $\text{The temperature was cooler than } 80^{\circ} \text{F and it was not sunny.}$
2. B) $\text{The temperature was cooler than } 80^{\circ} \text{F or it was not sunny.}$
3. C) $\text{If the temperature was at least } 80^{\circ} \text{F, then it was sunny.}$
4. D) $\text{If the temperature was cooler than } 80^{\circ} \text{F, then it was sunny.}$
5. E) $\text{If the temperature was cooler than } 80^{\circ} \text{F, then it was not sunny.}$
12.2022 AMC 10A #12
On Halloween, $31$ children walked into the principal's office asking for candy. They can be classified into three types: Some always lie; some always tell the truth; and some alternately lie and tell the truth. The alternaters arbitrarily choose their first response, either a lie or the truth, but each subsequent statement has the opposite truth value from its predecessor. The principal asked everyone the same three questions in this order.
"Are you a truth-teller?" The principal gave a piece of candy to each of the $22$ children who answered yes.
"Are you an alternater?" The principal gave a piece of candy to each of the $15$ children who answered yes.
"Are you a liar?" The principal gave a piece of candy to each of the $9$ children who answered yes.
How many pieces of candy in all did the principal give to the children who always tell the truth?
1. A) $7$
2. B) $12$
3. C) $21$
4. D) $27$
5. E) $31$
13.2021 AMC 10B #17
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is given $2$ cards out of a set of $10$ cards numbered $1,2,3, \dots,10.$ The score of a player is the sum of the numbers of their cards. The scores of the players are as follows: Ravon-- $11,$ Oscar-- $4,$ Aditi-- $7,$ Tyrone-- $16,$ Kim-- $17.$ Which of the following statements is true?
1. A) $\text{Ravon was given card 3.}$
2. B) $\text{Aditi was given card 3.}$
3. C) $\text{Ravon was given card 4.}$
4. D) $\text{Aditi was given card 4.}$
5. E) $\text{Tyrone was given card 7.}$
14.2016 AMC 10A #13
Five friends sat in a movie theater in a row containing $5$ seats, numbered $1$ to $5$ from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
15.2010 AMC 10A #15
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs?
1. A) $0$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
16.2001 AMC 10 #13
A telephone number has the form $\text{ABC-DEF-GHIJ}$ , where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$ , $D > E > F$ , and $G > H > I > J$ . Furthermore, $D$ , $E$ , and $F$ are consecutive even digits; $G$ , $H$ , $I$ , and $J$ are consecutive odd digits; and $A + B + C = 9$ . Find $A$ .
1. A) $4$
2. B) $5$
3. C) $6$
4. D) $7$
5. E) $8$
17.2000 AMC 10 #21
If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?
$\textrm{I. All alligators are creepy crawlers.}$ $\textrm{II. Some ferocious creatures are creepy crawlers.}$ $\textrm{III. Some alligators are not creepy crawlers.}$
1. A) $\text{I only}$
2. B) $\text{II only}$
3. C) $\text{III only}$
4. D) $\text{II and III only}$
5. E) $\text{None must be true}$
18.2003 AMC 10A #24
Sally has five red cards numbered $1$ through $5$ and four blue cards numbered $3$ through $6$ . She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards?
1. A) $8$
2. B) $9$
3. C) $10$
4. D) $11$
5. E) $12$

Logic Puzzles — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	E	To break an if-then statement, the if part must be true and the then part false. So n and n minus 2 must both be not prime.
2	B	All right means an A. So no A means not all right: at least one answer was wrong. That flipped-and-reversed statement is the only one that must be true.
3	B	Only Marta's place is given as an exact number, 6th. Start there and follow the clues one by one: Jack 5th, Todd 3rd, Rand 2nd, Hikmet 8th.
4	D	The facts stack up like circles inside circles: Crups inside Dramps inside Arogs inside Brafs. So every Crup is also an Arog and a Braf.
5	B	Every statement only talks about how many statements are true. Try each possible count from 0 to 4 and keep the one that agrees with itself.
6	B	The statement says a school cannot be both bigger than Euclid and a better seller than Euclid. Saying it in the other order gives the same meaning.
7	D	Follow the chain: a purple snake can't subtract, so it can't add. Every happy snake can add, so no happy snake can be purple.
8	A	Alice's doubt rules out 1 and 40. Then Bob can only be sure with 1, 2, 39 or 40; prime means 2, and 200 + 25 = 225.
9	E	Tuesday must be a running day, since four days in a row hold at most two runs. The tennis rule then rules out all but one schedule.
10	D	The 5-year-old is home, so the movie pair is 3 and 13 or 7 and 9. Only 3 and 13 leaves two boys under 10 for baseball.
11	B	Hot and sunny always means crowded. So if the beach was not crowded, the day was not both hot and sunny: at least one of the two was missing.
12	A	Write down how each kind of child answers the three questions. Truth-tellers say yes, no, no; liars say yes, yes, no; alternaters say no, no, no or yes, yes, yes.
13	C	Start with the smallest score. 4 can only be 1 + 3. Then 7 must be 2 + 5, and 11 must be 4 + 7.
14	B	The seat numbers always add to 15, so all moves cancel. Bea +2, Ceci -1, the swap 0: Ada moved one seat left.
15	D	Chris and LeRoy call each other frogs, so exactly one is a toad. Nobody could say Brian's sentence if Mike were a toad, so Mike is a frog.
16	E	Ten letters and ten digits means every digit 0 to 9 is used once. Pick the odd run and the even run; A, B, C are the leftovers.
17	B	Only statement II must be true: a creepy crawler that is an alligator is also ferocious. Statements I and III each fail in a small made-up example.
18	E	Red 5 fits beside only blue 5, and red 4 beside only blue 4. So those reds go at the two ends, and every other card is forced.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.