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?
- Teaching Elementary Probability [10/07/1998]
If you toss a number cube 20 times, could it land on six 20 times?
- Ten Card Arrangement Puzzle [10/04/2005]
Arrange ten cards numbered 1-10 in a pile. Turn over the top card,
then move the next card to the bottom of the pile. Turn over the new
top card and move the next card to the bottom of the pile. Continue
like this until all ten cards have been turned over. The challenge is
to arrange the pile so the cards are turned over in order from 1 to 10.
- Terms and Rules [12/14/2001]
My daughter is in sixth grade and has been doing a pattern journal where
she has two columns of numbers; the first column is the n and the second
column is the term, and she has to find the rule.
- Thinking about Number Pattern Problems [08/09/2004]
How do I figure out the next 2 numbers in the pattern 1, 8, 27, 64,
- Thinking about the Maximum Area Enclosed by a Fence [04/15/2004]
You have 2000 meters of fencing. What is the largest area you can
enclose with it using various shapes?
- 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.