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?
- Passing a Larger Cube Through a Smaller One [09/25/2003]
How is it possible to cut a hole through a solid cube so that a cube,
larger than the original, can be passed in one end and out the other?
- Passing Ships [05/12/1999]
Every day a ship leaves San Francisco for Tokyo... How many Tokyo ships
will each San Francisco ship meet?
- Perilous Ping-Pong [11/14/1997]
Weigh the balls to find the one that's different...
- The Pharaoh's Will [05/16/2000]
As he lay dying, the Pharaoh proclaimed: "I bequeath 1/3 of my estate to
my oldest child; 1/4 of my estate to the next oldest child; and to each
succeeding child, except the youngest, the next unit fraction of my
estate; and to the youngest the remainder."
- Placing Coins That Touch [8/7/1996]
How many 20-cent coins can you put around a 20-cent coin so that all of
- Planting Trees [08/13/2002]
I have to plant 10 trees in 5 rows with 4 trees in each row.
- Pool Table Algebra [10/21/1998]
The y-axis, x-axis, x = 6, and y = 12 determine the sides of a pool
table. Follow the path of a ball starting at the point (3,8).
- Positive Unit Fractions [10/02/2002]
Find five different positive unit fractions whose sum is 1. (A unit
fraction is a fraction whose numerator is 1. All denominators must
also be natural numbers.)
- Powers of Two [05/29/1997]
Prove that every power of two has a multiple whose decimal expansion has
only digits 1 and 2.
- A Practical Use for the Orthocenter [03/07/2001]
Does the orthocenter of a triangle have any practical uses?
- The Predetermined Sum Puzzle [02/02/2001]
I pick 5 digits, and write them down. My friend tells me a sum. Then I
pick 5 more digits, he picks 5, I pick 5, and he picks 5. The sum he told
me is the sum of all 5 lines. How did he know what it would be?
- Preparing for the Putnam Exam [11/04/2004]
I was wondering if you could give me some specific advice on how to
study for the Putnam Exam because I will be taking it for my first time.
- Prices in Store 88 [07/05/1999]
In Store 88 they sell exactly ten items, some items with the same price
as others, but the only the digit on the price tags of each item is 8...
- Prime Factors and Square Products [10/05/2003]
What is the smallest number that you can multiply by 540 to make a
- Primes and Number Riddles [4/15/1996]
A prime number riddle.
- The Prisoners' Dilemma [12/8/1995]
I'm looking for a paper - or some material - about "the prisoners'
- Product and Sum of Digits = Number [10/24/2001]
How many two-digit numbers exist such that when the products of their
digits are added to the sums of their digits, the result is equal to the
original two-digit number?
- Puzzle to Find a 10 Digit Number [10/10/2004]
I have to find a 10 digit number which uses each of the digits 0-9
such that the first digit is divisible by 1, the first two digits make
a number divisible by 2, the first three digits make a number
divisible by 3, and so on up to all ten digits making a number
divisible by 10. I figured it out using mostly guess and check, but
it took a long time. Is there a quicker way?
- Quickly Finding the Day of the Week [11/14/2000]
Today is November 14, 2000, a Tuesday. What day of the week was November
- Rameses' Pyramid [12/10/1997]
A pyramid-building puzzle.
- Rational Number and its Reciprocal [03/14/2002]
A rational number greater than one and its reciprocal have a sum of 2
1/6. What is this number? Express your answer as an improper fraction in
- Rectangles on a Chessboard [02/09/2002]
How many rectangles are there on a chessboard?
- Red Hat, Blue Hat [11/16/2002]
A teacher puts one hat on each of three students' heads and then
discards the remaining two hats so they cannot be seen. Then the first
child is told he can look at the other two children and from the color
of their hats, he can guess what color he is wearing...
- Rubik's Cube [12/03/1998]
Can you explain some of the math behind the Rubik's Cube?
- Rubik's Cube [02/18/1999]
Sequences of turns for solving Rubik's cube.
- Rubik's Cube Combinations [04/11/2001]
I read that a rubics cube has 4 quintillion different possible
combinations. Is this number correct? How can I calculate this value on
- Running Laps [07/03/2002]
Two dogs run around a circular track at different speeds. How long
will it take for them to return to the starting point at the same
- Russian Nim [02/15/1999]
Strategies for winning at Russian Nim (the "20" game).
- Scoring System Problem [10/28/2001]
What is the highest score that is impossible to make?
- Sequence Question from IQ Test [06/28/2007]
Find the missing number in the sequence 11 > ? > 1045 > 10445.
- Set Notation via Broken Typewriter [11/19/2002]
I have a machine that only types out ones (no spaces or tricks
involved). What procedure must you do to this machine to get any given
finite set? For example [2,85,11,5,60]. For a different set, the
number of ones that the machine types out will vary.
- Seven Elevators Stop at Six Floors [09/14/2002]
A building has 7 elevators, each stopping on at most 6 floors. If you
take the right elevator you can get to any floor from any other floor
without changing elevators. What is the greatest number of floors the
building can have?
- Shuffling Cards [8/26/1996]
How many 'perfect shuffles' does it take to get the cards back in the
order you started?
- Simultaneous Equations with Integral Solutions [11/29/1996]
What kind of a math project could I do with magic squares?
- Six-Card Trick [10/25/2000]
A Volunteer writes different numbers from 1 to 125 on six cards, and
keeps one. The Host arranges the others in some order and gives them to
the Partner, who then says the number on the missing card. How?
- Snail! [06/20/2002]
A snail is climbing a window-pane, beginning in the evening at a
height of e minus 1 meter from the base. It loses 1 meter each night.
On the second day, it doubles its altitude of the morning. On the
third day, it triples the altitude of the morning, and so on. What
will be its altitude on the 51st day at dawn?
- Solving a Math Poem [05/24/2000]
Take five times which plus half of what, and make the square of what
- Solving Questions [08/28/2002]
In a poll of 34 students, 16 felt confident solving quantitative
comparison questions, 20 felt confident solving multiple choice
questions.... How many students felt confident solving only multiple
choice questions and no others?
- Solving SEND + MORE = MONEY [04/18/2002]
I have tried logical reasoning and can't get it.
- A Special Ten-digit Number [02/17/1999]
Create a ten-digit number that meets some special conditions...