Ask Dr. Math

Middle School Archive

Browse Middle School Puzzles
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to frequently posed puzzles:
    Letter+number puzzles.
    Number sentences.
    Remainder/divisibility puzzles.

    1000 lockers.
    Getting across the river.
    Heads, legs: how many animals?
    A hen and a half...
    How many handshakes?
    Last one at the table.
    Measuring with two containers.
    Monkeys dividing coconuts.
    Squares in a checkerboard.
    Weighing a counterfeit coin.
    What color is my hat?

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Crossing a River: Cannibals and Missionaries [10/15/2001]

Three missionaries and three cannibals wish to cross a river using a single boat that will hold only two people. If at any time the cannibals outnumber the missionaries, those missionaries will be eaten...

Crossing the Desert [05/22/2001]

A truck gets one mile per gallon, and can hold 400 gallons at a time. How much is the minimum amount to cross a 1000-mile desert?

Cryptarithm with a Repeating Decimal [11/25/2004]

If FOCE / VOLVO = 0.COFFEECOFFEE....., find the value of COFFEE. I need to find a digit to replace each letter so that the pattern works, so there must be a fraction that leads to a repeating six-digit decimal.

Cryptography without Numbers [12/07/2003]

You want to send a valuable object to a friend. You have a box and several locks with keys. But your friend does not have the key to any lock that you have, and any key you send might be copied. How can you send to object safely?

Crypto-Number Puzzle [01/21/1997]

A Pascal program to find numerical values for the letters in: ONE + TWO + TWO + THREE + THREE = ELEVEN.

Cutting a Cake into 8 Pieces with 3 Cuts [09/08/2002]

How do you cut a cake into 8 pieces making only 3 cuts? How do you cut a doughnut into 12 pieces with only 3 cuts?

Cutting Slits in a Piece of Paper [02/01/2002]

How can you cut a hole in a piece of paper so a person can go through it?

Day Finding [07/17/1997]

What is the formula to find the day of the week, given the date, month, and year?

Decals on Lockers [02/28/2002]

The lockers in Pythagoras Middle School are numbered 1 through 500. Starting the count with locker 1, every 6th locker has a blue decal, evey 9th locker has a yellow decal and every 10th locker has a green decal. How many lockers have all 3 decals?

Digit Reversal Trick Explained [03/23/2001]

Take a 3-digit number and subtract its reverse. Then, take the result and add its reverse. Why is the answer is always 1089, no matter what the initial numbers were?

Dimes and Quarters Puzzle [11/5/1994]

Jessica has 16 dimes and quarters. Whitney has twice as many dimes and 1/3 as many quarters as Jessica has. They both have the same amount of money. What coins does each have?

Distance between Points on a Line [10/02/2002]

When the 10 distances between 5 pairs of points on a line are listed from smallest to largest, the list reads: 2,4,5,7,8,k,13,15,17,19. What is the value of k?

Dividing A Cake - A Math Puzzle [12/20/1998]

How can you cut a 9" square two-layer cake into 13 pieces so that each piece has exactly the same amount of cake and frosting?

Dividing a Clock into Three Equal Parts [10/27/2002]

Can you divide this clock into three parts using two lines and making all of the parts add up to the same number?

Dividing the Cost of a Hotel Room [11/20/1995]

A student tells a story about the division of the cost of the hotel room, but the charge doesn't seem to add up. Where did the extra dollar go? (See 'Tricky Word Problem', below.)

Doubling Pennies [11/26/1996]

If I start with a penny and double it daily for 30 days, how many pennies do I have at the end?

Doubling Pennies for a Year [06/03/2001]

How many pennies will we have at the end of 365 days if we begin on January 1 and double our money each day?

Duration of a Clock's Strike [01/21/2004]

If it takes seven seconds for a clock to strike 7 o'clock, then how long does it take to strike 10 o'clock?

Egg Timers: Use 7 and 11 to Make 15 [09/20/2001]

If you have only 7-minute and 11-minute hourglass egg timers, how can you time the cooking of an egg for 15 minutes?

Eight Circles, Sum is 13 [09/25/2002]

Arrange the numbers 1,2,3,4,5,6,7,8 in a given pattern so that the sum of the numbers in any line equals 13.

Eleven Sections of Fence [05/29/2003]

I have a grid containing 16 squares. Each square represents 1 acre. I need to draw 11 sections of fence along the dotted lines of the squares so that 4 fields are formed, each containing 4 acres of land.

Employee Scheduling [09/22/1998]

Can you help me make a schedule to staff an ice cream parlor?

Enumeration in Combinatorial Analysis [11/28/1996]

How many 3 digit numbers have the digital sum of nine?

Euclid's Algorithm for Finding a GCD [10/12/2002]

Find the only number besides one that is a common factor of 163231, 152057, and 135749.

Even-Numbered Magic Squares [04/30/2001]

How can I construct magic squares that work for even numbers?

Exchanging Seats in a Boat [06/08/2001]

Ten women are fishing in a long, narrow boat. One seat in the center of the boat is empty. The five women in the front of the boat want to change seats with the five women in the back of the boat...

Farmer, Goat, Wolf, and Cabbage Cross a River [03/13/1998]

Fewest number of trips a farmer can make to get a goat, wolf, and cabbage across a river.

A Fibonacci Jigsaw Puzzle [04/29/1999]

Why is the area of our rectangle, formed from a square, 65 when the square's area was 64?

Finding a Rule That Fits an Input-Output Problem [08/26/2006]

My daughter has an input-output chart problem where she has to find a rule and fill in the missing numbers. How can I help her with it?

Finding a Sequence of Numbers [08/13/2002]

Given six digits in a sequence: two 4's, two 5's, and two 6's, with one digit between the two 4's, two digits between the two 5's; and three digits between the two 6's, write this sequence of numbers.

Finding Input-Output Rules [10/08/2003]

My 3rd grader is having trouble finding rules to change one list of numbers (the inputs) into another (the outputs)...

Finding Magic Squares [05/12/1998]

Is there a pattern in solving magic squares?

Finding the Bad Machine by Gumball Weight [05/13/2008]

Nine gum ball machines produce gum balls that weigh 1 ounce, and one bad machine produces identical looking balls that weigh 0.5 ounce. With one weighing on a scale, determine which machine is defective.

Finding the Counterfeit Coin (1 of 9) [05/29/1998]

One of nine coins is counterfeit and is lighter or heavier than the others. How do I locate it in three weighings only?

Finding the Digits of SEND + MORE = MONEY [05/19/1998]

Representing and solving SEND + MORE = MONEY as a system of equations and inequalities.

Finding the Next Number in a Series [07/22/2002]

Are there any formal or systematic methods for solving problems that ask you to find the next number in a series?

Find Pairs of Positive Integers [10/15/2001]

Find pairs of positive integers whose greatest common factor is 1 and whose sum is 2000.

Find the 3-Digit Values A, B and C [08/08/1999]

How can I determine the 3-digit integers a, b, and c when the ratio a:b:c is 1:3:5 and the digits of a, b, and c are 1,2,3,...,9, each appearing exactly once?

Find the Heavier Coin of 12 in 3 Weighings [11/17/1996]

If there are twelve coins in a bag, one of which is either lighter or heavier than the others, find the fake coin in exactly 3 weighings AND determine if it is lighter or heavier than the others.

Find the Pattern, Complete the Combinations [01/13/2002]

Find the pattern: 3*4=5, 8*4=0, 3*7=2, 1*2=9, and use it to complete the combinations: 5*5=?, 4*4=?, 5*7=?

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