Math Forum - Problems Library - Pre-Algebra, Logic & Set Theory: Logical Reasoning

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Logical Reasoning
Problems in this section require students to formulate conjectures and to assess them on the basis of evidence.

Related Resources
Interactive resources from our Math Tools project:
   Math 7: Logic & Set Theory
The closest match in our Ask Dr. Math archives:
   Middle School: Logic
NCTM Standards:
   Reasoning and Proof Standard for Grades 6-8

Access to these problems requires a Membership.

Accounting for Age - Suzanne Alejandre
Pre-Algebra, difficulty level 3. Given a number trick, explain why the result is always the original number followed by your age. ... more>>

Anjelica's Very Odd Magic Sweater - Cynthia Lanius, Suzanne Alejandre
Pre-Algebra, difficulty level 3. Find the shortcut Angelica's mother might have used to calculate the number of dimes. ... more>>

Balancing Mobiles - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Explore two Laws of Archimedes as you puzzle through a balanced mobile. ... more>>

Baseball Attendance Figures - Judy Ann Brown
Pre-Algebra, difficulty level 3. Find the difference between the greatest possible number of people at the stadium and the least possible number of paid admissions to the game. ... more>>

Birthday Line Up - Claire Mead, Stephen Weimar
Pre-Algebra, difficulty level 2. How old will Seth be when his mother is five times Seth's age? ... more>>

Boxes of Stationery - Annie Fetter, Suzanne Alejandre
Pre-Algebra, difficulty level 3. How many sheets of paper and how many envelopes are in the box of stationery? ... more>>

Caliyah's Cart - Lillian Ray
Pre-Algebra, difficulty level 4. How many items did Caliyah decide to buy at the store? ... more>>

Car Rental Quandary - Judy Ann Brown
Pre-Algebra, difficulty level 3. Compare the car rental deals using the given information. ... more>>

Club Competition - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Determine how many teams the Fibonacci Middle School Math Club can enter in their first competition. ... more>>

Coin Count - Roya Salehi and Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the given information to determine the number of nickels and dimes in Lisa's bank. ... more>>

Commuting Conundrum - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Help Suzanne and Richard decide which train tickets will give them the most value for their dollar. ... more>>

A Cranberry Craving - Claire Mead, Suzanne Alejandre
Pre-Algebra, difficulty level 2. Use the pattern to find out how many cranberries Carissa ate on Thanksgiving Thursday. ... more>>

Duke is Missing - Judy Ann Brown
Pre-Algebra, difficulty level 3. Duke's head is 6 inches long, his tail is as long as his head plus half the length of his body, and his body is as long as the head and tail combined. How long is Duke? ... more>>

Eight Squares on a Table - Suzanne Alejandre
Pre-Algebra, difficulty level 3. Find the number written on the first sheet laid on the table and identify which of the eight sheets it was. ... more>>

The Graduating Class - Judy Ann Brown
Pre-Algebra, difficulty level 3. Use the information about students with different grades to determine the number of students in Mrs. Smith's class. ... more>>

Guess My Number If You Can! - Judy Ann Brown
Pre-Algebra, difficulty level 4. Use division clues to reveal the secret number. ... more>>

Happy New Year! - Judy Ann Brown
Pre-Algebra, difficulty level 4. A puzzle for the new year: how is the grid designed? ... more>>

Happy Thanksgiving - Judy Ann Brown
Pre-Algebra, difficulty level 3. A logic puzzle to solve the age-old question, "Who will wash the dishes?" ... more>>

How Old is Ms. Nelson? - Judy Ann Brown
Pre-Algebra, difficulty level 2. Ms. Nelson claims that she is three times as old as her little sister. However, six years ago she was five times as old as her sister was. How old is Ms. Nelson? ... more>>

It's A Secret - Judy Ann Brown
Pre-Algebra, difficulty level 2. In this puzzle, your job is to decode the secret message. ... more>>

Label Shuffle - Judy Ann Brown
Pre-Algebra, difficulty level 3. There are three containers. Which one has the red marbles? ... more>>

Lemonade Fun - Maria Fung and her students
Pre-Algebra, difficulty level 2. How many cups of each size lemonade did Jose and Nick sell? ... more>>

Locker Combinations - Judy Ann Brown
Pre-Algebra, difficulty level 4. In this counting problem you will need to determine the total number of possible locker combinations. ... more>>

Lost in Des Moines - Judy Ann Brown
Pre-Algebra, difficulty level 2. Calculate the speed of a car using given information. ... more>>

Making Money - Alice Miller
Pre-Algebra, difficulty level 3. How much money would you have by December 31, 1999, if you put 10 cents in the special "math" bank on January 1, 1999? ... more>>

Math Club Lollipop Profits - Judy Ann Brown
Math Fundamentals, difficulty level 3. The Math Club plans to sell lollipops. Help them to calculate their profit. ... more>>

Math Club Ski Trip - Judy Ann Brown
Pre-Algebra, difficulty level 2. Mrs. Germain and her friend, Mrs. Granville, need your help to plan the shortest route to visit five local ski areas. ... more>>

Measuring Melons - Suzanne Alejandre, Claire Mead
Pre-Algebra, difficulty level 3. Students determine how much each type of fruit weighs. ... more>>

Movie Seating - Maria Fung and her students
Pre-Algebra, difficulty level 2. Use the clues to determine where Mom, Dad, Randy, Shelley, Noreen, and Alan sit at the movie theater. ... more>>

Mr. Carmel's Party Plan - Judy Ann Brown
Pre-Algebra, difficulty level 2. Help Mr. Carmel's class earn an ice cream party by counting the number of possible sundaes. ... more>>

The Name Game - Judy Ann Brown
Pre-Algebra, difficulty level 3. The first student says his/her own name; the second student repeats the first student's name and adds his/her own name... How many names are said if the numbers of students in my math classes on the first day are 27, 28, 32, 31, and 30? ... more>>

Operating Diagonally - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Think about the effect of multiplication and addition on the numbers in the hundreds chart. ... more>>

The Oracle's Crowns - Suzanne Alejandre
Pre-Algebra, difficulty level 3. Figure out which crown Caleb had on and how he knew for sure. ... more>>

The Oracle's Oration - Annie Fetter
Pre-Algebra, difficulty level 3. Explain how payment should be distributed based on the number of rolls two people shared with the group of three. ... more>>

Ostrich Llama Count - Suzanne Alejandre
Pre-Algebra, difficulty level 2. Given the combined number of heads and legs of ostriches and llamas, help Raul and Esteban figure out how many ostriches and how many llamas there are. ... more>>

Peanut's Back Yard - Jonathan Bauer
Pre-Algebra, difficulty level 3. Peanut, the overexcited dog, makes 40 laps around a stake and travels a total distance of 1 mile. If his bowl of water is 20 feet away from the stake, can he get a drink? ... more>>

Pegging Possibilities - Michael Urbut
Pre-Algebra, difficulty level 2. Explore the number of possible arrangements of pegs in this logic game. ... more>>

Pouring the Punch - Neil Bress
Pre-Algebra, difficulty level 3. The Refreshments Committee needs to pour exactly four liters of eggnog using only three-liter and five-liter measures. ... more>>

Power Outage - Jackson Geometry Class
Pre-Algebra, difficulty level 3. The power went out. Use the current reading on the clocks to determine when the power went out, and for how long. ... more>>

Pumpkin Carving for Charity - Suzanne Alejandre
Pre-Algebra, difficulty level 1. How much money did each of the six friends raise by carving pumpkins to auction off at a local charity event? ... more>>

Ready for Football? - Terry Trotter & Suzanne Alejandre
Pre-Algebra, difficulty level 2. Find the final score of the game between the Eagles and the Dragons. ... more>>

Runsums, Part 1 - Ron Knott
Pre-Algebra, difficulty level 3. By spotting the pattern, tell me the smallest number that is a sum of a run of one hundred consecutive numbers. ... more>>

Runsums, Part 2 - Ron Knott
Pre-Algebra, difficulty level 3. Explore number patterns and report on what you have learned. ... more>>

Santa's Little Helpers - Judy Ann Brown
Pre-Algebra, difficulty level 3. Help Santa to solve the puzzle created by the clever elves. ... more>>

Seating Dilemma - Judy Ann Brown
Pre-Algebra, difficulty level 2. Count the number of ways to seat people, if some are math teachers who cannot be next to each other. ... more>>

A Simple Reversal - Judy Ann Brown
Math Fundamentals, difficulty level 1. The product of 3 and 24 is 72. The sum of 3 and 24 is 27, which is the reverse of 72. Can you find other pairs of numbers such that their product is the reverse of their sum? ... more>>

Snow Day Phone Chain - Judy Ann Brown
Pre-Algebra, difficulty level 4. Use the given information to find out how many phone calls were made. ... more>>

A Snowy Driveway - Claire Mead
Pre-Algebra, difficulty level 3. Usually Bertha and Gertrude shovel snow together. How long would it take Bertha to shovel the same amount alone? ... more>>

Square Dancing - Judy Ann Brown
Pre-Algebra, difficulty level 2. The sum of each couple's numbers is a perfect square. Use other information in the problem to find the number of Ashley's partner. ... more>>

Top Notch Nose Contest* - Judy Ann Brown, Suzanne Alejandre
Pre-Algebra, difficulty level 2. Use the given information to determine how many photos of pets were entered in the contest. ... more>>

The Tower of Hanoi Puzzle - Roya Salehi
Pre-Algebra, difficulty level 4. Tower of Hanoi: What is the minimum number of moves it will take to transfer 25 disks from one post to another? ... more>>

Train Travel - Annie Fetter & Ian Underwood
Pre-Algebra, difficulty level 2. At what time does the end of the train emerge from the tunnel? ... more>>

Tunnel Time - Annie Fetter, Ian Underwood, Suzanne Alejandre
Pre-Algebra, difficulty level 2. At what time does the end of the train emerge from the tunnel? ... more>>

Two Heads Are Better Than One - Lillian Ray
Pre-Algebra, difficulty level 3. Given the combined number of heads and legs of ostriches and pushmi-pullyus, will Raquel and Esperanza be able to know for sure how many ostriches and pushmi-pullyus are on the farm? ... more>>

What Day Is It Today? - Roya Salehi
Math Fundamentals, difficulty level 2. Sophie tells the truth only on Monday, Tuesday, Wednesday, and Thursday. Her sister tells the truth only on Monday, Friday, Saturday, and Sunday. If they both confess to their mom that "Yesterday I lied," what day is it today? ... more>>

Wilda's Swamp Water - Judy Ann Brown
Pre-Algebra, difficulty level 3. Measure 8 quibits of swamp water using only 6-quibit and 10-quibit containers. ... more>>

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