See also the
Dr. Math FAQ:
3D and higher
Browse Middle School Puzzles
Stars indicate particularly interesting answers or
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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?
- Thinking Outside the Pen [09/28/2004]
How can you put 21 pigs in 4 pig pens and still have an odd number of
pigs in each pen?
- Thoughtful Trial and Error [05/21/2002]
Arrange the numbers 1 through 10 in a triangle so that all the rows of
three numbers have the same sum, and all the rows of four numbers have
the same sum.
- Three Cuts and Seven Pieces [01/18/1999]
Is there any way that you can cut a pie into seven pieces with just three
- Three Holes Puzzle [05/02/2002]
A piece of plywood has three holes it it: a circular hole with a
diameter of 2 cm, a square hole with 2 cm sides, and a triangular hole
with a base and height of 2 cm. What object could completely plug AND
pass completely through each hole?
- Three Jars [11/01/2000]
One jar has red marbles, one has black, and one has both, but their
labels are wrong. Can you fix the labels by picking one marble from one
- Three Weights [12/07/1997]
A boy selling fruits has only three weights, but with them he can weigh
any whole number of pounds from 1 pound to 13 pounds inclusive. What
three weights does he have?
- To Find the Heavy Marble, Divide and Conquer Asymmetrically [01/14/2011]
A student knows how to deduce the one heavy marble out of eight by weighing
successive subsets on a scale balance three times ... but after only two weighings?
Doctor Ali suggests the first step.
- Toothpick Puzzle [09/27/2004]
If you have a square that is made up of nine little squares, each one
toothpick per side, so the big square is three toothpicks per side,
can you remove 5 toothpicks and leave 3 squares (of any size)?
- Tower of Hanoi [10/16/1997]
I'm looking for a mathematical solution, not a trial-and-error one.
- Towers of Hanoi Puzzle: 3 Pegs, n Discs... [11/15/1997]
We are told to find out the least number of moves it takes to get three
discs in size order onto the third peg from the first.
- Tracing a Figure Without Lifting Your Pencil [03/09/2001]
Is there a simple way to quickly tell whether a figure can be traced
without lifting your pencil?
- Trains and Fractions [12/24/1994]
A non-stop train leaves city A for city B at 60 m.p.h.... If to the
numerator and denominator of the fraction 1/3 you add its denominator...
A train moving at 45 m.p.h. meets and is passed by a train moving at 36
- The Traveling Businessman Dilemma [04/09/1998]
Three businessmen are on a trip and need a place to stay for the night...
- Twenty or Fewer Steps [08/28/2002]
Only the 1, +, -, x, /, (, ), and = keys on a scientific calculator
are working. How can a result of 75 be reached by pushing these
keys fewer than 20 times?
- Two 2s Make 5 [01/03/2003]
Using only two 2s and any of the standard mathematical symbols, write
an expression whose value is equal to exactly five.
- Two Mathematicians: Factoring Logic [03/24/2003]
Two mathematicians are each assigned a positive integer. They are told
that the product of the two numbers is either 8 or 16. Neither knows
the other's number...
- Two Mathematicians Problem [05/18/1998]
One mathematician is give the sum of integers X and Y, and another is
given their product... what are the numbers?
- Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]
Find two numbers such that when you find the sum, product, and
quotient of the two numbers they are all equal.
- Two Quantities, Two Relations [04/04/2000]
Find two consecutive numbers whose squares, when subtracted, equal 43.
- Unit Fractions and the Greedy Algorithm [12/27/2000]
How can I represent 2000/2001 as the sum of unit fractions?
- Unit Fractions Summing to 1 [07/15/2001]
Find seven different unit fractions whose sum is 1.
- Units Digit of 2^57 [10/07/2001]
What is the units digit of 2 to the power of 57?
- U.S. and European Sock Sizes [03/23/2002]
Which expression could be used to convert European size to U.S. size?
- Use 10 Digits in 2 Fractions that Add to 1 [05/26/2003]
Obtain a sum of the form xx/xxx + xx/xxx = 1, using all digits 0-9
- Use 2,3,7,8 to Write Math Expressions from 1 - 50 [10/03/2004]
Using each of the digits 2,3,7,8 one time, and any math operations you
know, find expressions with values of 44 and 46. I have done from 1
to 50 except these last two numbers.
- Use One Weighing [02/06/2002]
There are 90 genuine coins and 10 fake coins in 10 stacks with 10 coins
in each stack... Using only one weighing, determine which stack contains
the fake coins.
- Using Prime Factors to Limit Search [07/26/2002]
Our courtyard has more than one tree, and each tree contains more than
- The Value of a Word [08/25/1998]
Think of a word that equals one dollar. The key is: a=.01, b=.02, c=.03,
- Venn Diagram: Goops, Gorps, Gorgs [09/19/2002]
Every Goop is a Gorp. Half of all Gorgs are Gorps. Half of all Gorps
are Goops. There are 40 Gorgs and 30 Goops. No Gorg is a Goop. How
many Gorps are neither Goops nor Gorgs?
- Visualizing Two Word Problems [03/12/1998]
Sandy has thrice as many beads as Wendy... Paul gives 20 of his marbles
- Weighing Bolts [2/14/1996]
A hardware plant makes rivets 6 inches long and 6 ounces in weight. Harry
makes a drum of rivets 6 inches long weighing only 5 ounces each. Harry's
drum is stored along with 9 other proper drums. Harry has to find the
rivets but he can only use a scale one time, and can take only one
reading from that scale. How can Harry be sure of picking the right drum
out of the ten?
- Weight of Each Bale of Hay [09/26/1997]
Five bales of hay are weighed in all possible combinations of two...
- What's His Street Address? How Many Neighbors Does He Have? [09/29/2010]
A student struggles with a word problem that asks for specific sums of counting
numbers. Three different doctors weigh in with increasingly sophisticated and
comprehensive problem-solving approaches: programming spreadsheet formulas;
applying combinatorics; and invoking quadratic Diophantine and Pell equations.
- Where Do the Parentheses Go? [09/12/2003]
Add parentheses to 2 + 7^2 - 3^2 / 3^2 - 1 * 5 = 35 to make it a true
- Where to Put the Parentheses? [01/03/2002]
I am stuck on where to put parentheses in the math expression 9-6+4*6/
3=-2 to make the expression true.
- Which Car is in the Pole Position? [08/14/1999]
Clues for finding starting positions of the 10 cars entered in the
Rithmetic Road Rally.
- Which Grasshopper Wins the Race? [06/11/2003]
The larger grasshopper covers 10 inches per jump, while the smaller
one covers 5 inches. The smaller grasshopper takes 2 jumps for every
jump of the larger grasshopper. Which one wins, and why?
- Who Finished In What Place? [12/15/2002]
Sarah, Carla, and Kim were the finalists at a gymnastics meet...
- Who Made Which Toys? [12/21/1998]
A math logic problem, from a rhyme describing Santa's toymakers.
- Why the Motionless Runner Parodox Fails [01/01/2005]
I read about the Motionless Runner paradox on your site, and I am now
convinced that motion is an illusion. Can you help me understand why
the paradox can't be true?