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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A Billion Seconds [10/21/1996]

If I was born on 01/02/84, when will I have been alive for a billion seconds?

Binary Notation Card Trick [12/10/2002]

My 7th grade teacher had a card game that had about 7 cards. She said she could guess your number by asking you if your number was on each card. One card had even numbers and the other had odd.

Bingo [8/2/1995]

Is there any literature on the number of ways of winning at bingo? How many possible different bingo cards are there?

Birthdate Puzzle [01/11/1998]

The problem starts: write the number of the month in which you were born... it all works, but the teacher wants to know why.

Birthday Number Puzzle Variations [11/18/2002]

A friend showed me a number trick that works no matter what number I start with. How does it work? He also said that the year 2002 is the only year it will ever work. Is that true?

Borrowing and Returning Books [03/26/2001]

Four students borrowed books from a bookshelf and eight students returned books. There are 27 books on the shelf. How many books were there in the beginning?

Bowling Pin Problem [02/17/2006]

Ten bowling pins are set up in the usual way forming a triangle with the point facing the bowler. How can three pins be moved so that the point now faces away from the bowler?

Box A or Box B? [11/21/2000]

A professor shows you two boxes. Box A has $10K; Box B has $1M or $0. You can take Box B only or both boxes... Which do you choose?

Breaking One Big Problem into Several Smaller Ones [09/11/2003]

How can I change a dollar into exactly 26 coins?

Bulgarian Goats [05/17/2003]

How many goats are there in the herd? What are the sizes of the feeding groups once they have stabilised? Find at least two possible cyclic patterns of sizes.

Calendar Puzzle [12/02/1999]

A student cuts a 3x3 square from a calendar page. If the sum of the nine dates is divisible by 19, what is the date in the lower left-hand corner of the square?

Candy Bar Puzzle [10/28/1996]

Raymond gives candy bars away in a specific pattern until he has no candy bars left. How many candy bars did he start with?

Can the Sum of 5 Odd Numbers Equal an Even Number? [01/05/2003]

What are 5 odd numbers which, when added, equal 32?

The Case of the 90 Apples [11/19/2002]

A Cadi gives 90 apples to three girls and says: you must sell these apples in the market. Fatima, the eldest, takes 50, Cunda 30, and Siha, the youngest, the remainder (10). If Fatima sells 7 apples for 1 dinar the others will sell their apples at the same price. The three girls must sell all their apples and must get the same amount of money.

Challenging Algebra Age Problem [02/29/2004]

A man has nine children whose ages are at an exact interval. The sum of the squares of the ages of each is the square of his own age. What is the age of each child and the man?

Change for a Dollar [03/07/1999]

How many different ways are there to make a change for a dollar?

Change for a Dollar [11/05/1999]

When changing a dollar bill, you can give 1 coin (1 silver dollar), 2 coins (2 half-dollars), 3 coins (2 quarters and 1 half-dollar), and so on. What is the least positive number of coins that is impossible to give as change for a dollar bill?

Checkerboard Pattern Using an L Shape [10/14/2002]

Place the numbers 1-64 in L-shaped patterns on an 8x8 checkboard. Can you always get an L shape with the sum of the three digits as odd using up all the available squares on the board?

Chess Problem [11/16/1994]

We are investigating the number of squares on a chess board "controlled" by the Queen at any one position, i.e., where she could move (anywhere in a straight line).

Choosing 3 of 6 Colors [03/03/2000]

Patrick has a box of crayons with red, blue, yellow, orange, green and purple. How many different ways can Patrick select 3 colors?

Claim the Last Flag [10/25/2002]

Two teams face 21 flags. Teams take turns choosing 1, 2, or 3 flags at each turn. The team that can claim the last flag wins.

Clock Hands and Hours Worked [01/29/2003]

A man arrives at work between 8:00 and 9:00 a.m. at exactly the moment the minute hand and hour hand of the clock point in the same direction.... How many hours did the man work?

Clock Hands Diametrically Opposite [10/10/2001]

At what time between two and three o'clock are the hands of a clock diametrically opposite?

Coconut and Monkey Puzzle [12/18/1997]

How many coconuts were in the original pile?

Coconut Piles [10/12/1998]

What is the least number of coconuts they could have started with?

Coffee or Tea? [07/09/2001]

Is there more coffee in the tea, or more tea in the coffee, or are they the same?

Coin Patterns [08/26/2003]

Each of four rows of coins has exactly one penny, one nickel, one dime, and one quarter. No row, either horizontal, vertical, or diagonal, has more than one coin of each kind. How are the coins arranged?

Coins in Change under $1 [03/13/2001]

Is there a formula or equation for determining the smallest number of coins a person could receive when given change less than $1.00?

Combining Boxes [04/27/1998]

Uncle Dube's Dandelion Delectables come only in boxes of 6, 9 or 20. What is the largest number of Delectables you can't buy?

Combining Numbers [08/04/2002]

Combine the numbers 3, 3, 8, and 8, using only addition, subtraction, multiplication, and division, to come up with the number 24.

Comparing Sums of Odd and Even Integers [03/31/2004]

Which is greater, the sum of all of the even numbers that are less than 100, or the sum of all of the odd numbers that are less than 100? It takes too long to add them all up!

Concept of Farmer Crossing a River [04/30/2002]

What concept does the farmer, fox, and chicken problem use, and how does it relate to the sheriffs and outlaws problem, husband and wives problem, and missionaries and cannibals problem?

Connecting the Boxes [12/28/1998]

I have an arrangement of boxes and am trying to draw one continuous line connecting them all. Can this be done?

Connect Nine Dots [11/4/1996]

Draw four lines connecting nine dots in a grid without lifting the pen from the paper.

Constructing a Conditional Table [04/14/1998]

I have a table that is 5 columns wide and 7 rows high and contains either a 0 or a 1 in each cell...

Counterfeit Money [05/28/1999]

How can I find the amount of money a shoestore owner lost when he was passed a counterfeit $20 bill?

Counting Days of the Week [02/02/1999]

If today is Thursday, what day would it be 120 days from now?

Counting Rectangles Cut By a Diagonal [06/15/1999]

How can we find an equation for the number of unit squares that are cut by a line going from corner to corner on a rectangle?

Counting Seconds [12/13/2002]

Ali counted to 10, speaking a number every 5 seconds. How long did the complete count take?

Criminal Logic Problem [02/19/2005]

A challenging logic problem involving five criminals charged with five crimes. The names of the criminals are the same as the crimes, but no criminal commited the crime of his name. Using several clues, determine who committed murder.

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