See also the
Dr. Math FAQ:
Browse High School Puzzles
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to frequently posed puzzles:
Getting across the river.
How many handshakes?
Last one at the table.
Monkeys dividing coconuts.
Squares in a checkerboard.
Weighing a counterfeit coin.
What color is my hat?
- Twenty Quadrilaterals from Nine Dots [04/04/1999]
How can you get 20 quadrilaterals from 9 dots?
- 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.
- 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.
- 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
- Using Dissections to Cut and Reshape Rectangles [08/28/2004]
Cut a 20x15 piece of paper into two pieces to form a 25x12 piece.
- Using Prime Factors to Limit Search [07/26/2002]
Our courtyard has more than one tree, and each tree contains more than
- USSR Math Olympiad Puzzle [04/16/2003]
Prove that no matter what string you start with, the letters at
the corners of the triangle are either all the same or all different.
To what other numbers could you change the 'string of 10 letters' and
still have the assertion be true?
- 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?
- Venn Diagram: Two Possibilities [01/14/2003]
The science club advisor asked club members what science courses they
liked. Eighteen members said they liked physics, 17 liked chemistry,
and 10 liked biology. However, of these, 9 liked physics and
chemistry, 4 liked biology and chemistry, 2 liked physics and biology,
and 2 liked all three. How many science club members were interviewed?
- Version of Lights-Out Puzzle Using Buttons [09/02/2004]
A variation on the classic Lights-Out Puzzle in which pushing any
button in a 6x6 grid changes the state of that button and all others
in the same row or column. The goal is to maximize the number of
buttons in a given state.
- The Vicar and the Curate (Ages of Three People) [03/20/2002]
The product of the ages of three people is 2450 and the sum is twice the
age of the curate. How old are the three people?
- Walking the Shortest Distance [10/07/2002]
A single boy lives in each of n equally spaced houses on a straight
line. At what point should the boys meet so that the sum of the
distances that they walk from their houses is as small as possible?
- Ways to Make Change for $1.00 [10/02/1997]
Are there really 292 ways to make change for $1.00?
- A Way to Think about the Locker Problem [01/16/2004]
Imagine there is an endless string of lockers in your school. Person 1
starts at locker 1 and opens every locker. Person 2 starts at locker 2
and closes every 2nd locker. Person 3 starts at locker 3 and changes
every 3rd locker. Person 4 starts at locker 4 and changes every 4th
locker. Person x starts at locker x and changes every xth locker.
Which lockers are left open in rows of 25, 100, and 500 lockers?
- Weighing Bales of Hay [12/10/1997]
Five bales of hay are weighed in all possible combinations of two...
- Weight of Each Bale of Hay [09/26/1997]
Five bales of hay are weighed in all possible combinations of two...
- Weird Fraction Behavior? [11/4/1994]
If you look at the fractions (16/64) and (19/95), you may notice that if
you cancel out the second number in the numerator with the first number
in the denominator the fraction remaining is equivalent to that of the
original equation. Ex. in the fraction (16/64) if you cancel out the
second number in the numerator (6) with the first number in the
denominator (6), you end up with (1/4), which is equal to (16/64). The
only restrictions are that the numbers canceling must be the same number,
as in the above example (a 6 for a 6). Also the numbers for the original
fraction are restricted to two digits (10-99). How many more of these
numbers can you find?
- 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.
- When will Ramadan fall across 3 months? [3/5/1995]
This year the Muslim holy month of Ramadan fell across 3 months of the
western calendar. It began 31 Jan & ended 1 Mar....
- Where is the Arsenic? [03/12/2002]
You place six jars (right to left: coffee, arsenic, and sugar on the top
shelf; snuff, tea, and salt on the bottom shelf)...
- Who Got Engaged to Whom? [11/27/2001]
Dorothy, Jean, Virginia, Bill, Jim, and Tom became engaged to one
another. Who got engaged to whom?
- Who Made Which Toys? [12/21/1998]
A math logic problem, from a rhyme describing Santa's toymakers.
- Who Owns the Fish? (Einstein's Problem) [07/18/2002]
There are 5 houses sitting next to each other, each with a different
color, occupied by 5 guys from 5 different countries...
- 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?
- Wilbert the Wonder Dog [11/14/2002]
Five masters and their five dogs need to cross a river.
- Wile E. Coyote Lands in the River [08/20/1999]
Wile E. Coyote is standing on a springboard atop a high cliff. Road-
runner drops a boulder on the other end of the springboard, sending Wile
up at an initial velocity of 4 m/s. At what time will he land in the
river, 120 m below the top cliff?
- Winning at NIM [06/09/1997]
How do you ensure that you win the game of NIM?
- Winning at NIM [07/25/1998]
In a game of NIM, there are three rows of 5, 4, and 3 sticks
respectively. Picking up as many as you want in a row, how do you win?
- Words Equal to a Dollar [10/11/2001]
I need one-dollar words: a = 1, b = 2, c = 3...z = 26. I would like a
list of words that equal 100.
- World War II Window Blackout [10/21/2001]
Mr. Brown had a square window 120cm x 120cm, but the only material he
could find was a sheet of plywood 160cm x 90cm; same area, different
shape. He drew some lines and cut out just two congruent shapes, which he
joined to make a square of the correct size. How did he do it?
- Write a Sum That Totals 100 Using Digits 1 - 9 [05/30/2004]
Is it possible to arrange the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 so that when added they total 100? Only adding is permitted and the numbers can be rearranged. Each number can only be used once.
- Zeno's Paradox [10/19/1995]
At eleven o'clock I put ten balls numbered 1,2, ...10 in a box and
immediately take out the ball numbered 1. At eleven thirty I put balls
numbered 11 through 20 into the box and take out the ball numbered 2. At
eleven forty-five I put balls numbered 21 through 30 into the box and
take out the ball numbered 3. This continues at time intervals that are
half of the preceding one. How many balls are in the box at twelve