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

Logic, Statistics & Misc · 18 problems have this as their main idea

primary-topic problems (1.4% of all)

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)

Problems, easiest first

Open any of them to try it here; the worked solution is on the problem's own page.

2019 AMC 10B · #2Logic Puzzles

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?

2017 AMC 10A · #6Logic Puzzles

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?

2015 AMC 10B · #5Logic Puzzles

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 …

2007 AMC 10B · #5Logic Puzzles

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?

2025 AMC 10A · #8Logic Puzzles

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 …

2022 AMC 10B · #11Logic Puzzles

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"?

2021 AMC 10A · #7Logic Puzzles

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?

2021 AMC Fall 10B · #10Logic Puzzles

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 …

2015 AMC 10B · #6Logic Puzzles

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?

2013 AMC 10A · #6Logic Puzzles

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?

2011 AMC 10B · #8Logic Puzzles

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?

2022 AMC 10A · #12Logic Puzzles

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 …

2021 AMC 10B · #17Logic Puzzles

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 …

2016 AMC 10A · #13Logic Puzzles

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 …

2010 AMC 10A · #15Logic Puzzles

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: …

2001 AMC 10 · #13Logic Puzzles

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 …

2000 AMC 10 · #21Logic Puzzles

If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true? $\textrm{I. All alligators are creepy crawlers.}$ $\textrm{II. Some ferocious creatures are creepy crawlers.}$ $\textrm{III. Some alligators are not creepy crawlers.}$

2003 AMC 10A · #24Logic Puzzles

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?