Logic Puzzles

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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)$12$

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.