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?
- 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.
- 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,
- 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?
- Fifty-two Cards, Three Piles ... One Coincidence? [02/20/2013]
Why does the "count on" card trick work? Doctor Peterson revisits a card-counting
curiosity with greater clarity.
- 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
- 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