See also the
Dr. Math FAQ:
3D and higher
Browse Middle School Puzzles
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to frequently posed puzzles:
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?
- A Bidirectional Search [01/27/2003]
How can I get the numbers from 1-100 using only four 8's?
- A Billion Seconds [10/21/1996]
If I was born on 01/02/84, when will I have been alive for a billion
- 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
- 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
- The Case of the 90 Apples, Redux [05/14/2016]
A non-native speaker of English shares ill-posed puzzles about how selling different
quantities of the same goods results in identical profits. Doctors Ian and Peterson
compare it to a similar puzzle before providing insights general and specific about
tiered pricing schemes.
- 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
- 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
- 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
- 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?