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?
- Change for a Dollar [11/05/1999]
When changing a dollar bill, you can give 1 coin (1 silver dollar), 2
coins (2 half-dollars), 3 coins (2 quarters and 1 half-dollar), and so
on. What is the least positive number of coins that is impossible to give
as change for a dollar bill?
- Checkerboard Rectangles [03/24/2003]
A checkerboard has 8 horizontal boxes and 8 vertical boxes. How many
rectangles are possible inside that board?
- Choosing 3 of 6 Colors [03/03/2000]
Patrick has a box of crayons with red, blue, yellow, orange, green and
purple. How many different ways can Patrick select 3 colors?
- Circle Packing [01/22/2001]
If circles packed in a 100 by 100 square are repacked so that the centers
of any three tangent circles form an equilateral triangle, what is the
maximum number of additional circles that can be packed?
- A Circular Massacre [09/25/1998]
Ten thousand sailors are arranged in a circle; starting with the first
one, every other sailor is pushed overboard ....
- Claim the Last Flag [10/25/2002]
Two teams face 21 flags. Teams take turns choosing 1, 2, or 3 flags at
each turn. The team that can claim the last flag wins.
- Clock Hands and Hours Worked [01/29/2003]
A man arrives at work between 8:00 and 9:00 a.m. at exactly the
moment the minute hand and hour hand of the clock point in the same
direction.... How many hours did the man work?
- Clock Hands Diametrically Opposite [10/10/2001]
At what time between two and three o'clock are the hands of a clock
- Clock Hands Trisecting Face? [04/05/1999]
At what time (if any) do the three hands of a clock trisect its face?
- Closest Palindromic Dates [02/07/2000]
Using the abbrevation date.month.year, what are the two palindromic dates
closest together in the 1900s?
- Cockroach Traveling Along an Elastic Tightrope [05/15/1998]
Finding the harmonic series in a problem of related rates.
- Coconut and Monkey Puzzle [12/18/1997]
How many coconuts were in the original pile?
- Coconut Piles [10/12/1998]
What is the least number of coconuts they could have started with?
- Coconuts, Forwards and Backwards [02/02/2010]
Doctor Greenie answers a chestnut about repeated division and
remainders, first working the question forwards before using the
inverse of a function to solve the same problem backwards much more
- Coffee or Tea? [07/09/2001]
Is there more coffee in the tea, or more tea in the coffee, or are they
- Coin Patterns [08/26/2003]
Each of four rows of coins has exactly one penny, one nickel, one
dime, and one quarter. No row, either horizontal, vertical, or
diagonal, has more than one coin of each kind. How are the coins
- Coins in Change under $1 [03/13/2001]
Is there a formula or equation for determining the smallest number of
coins a person could receive when given change less than $1.00?
- Concept of Farmer Crossing a River [04/30/2002]
What concept does the farmer, fox, and chicken problem use, and how
does it relate to the sheriffs and outlaws problem, husband and wives
problem, and missionaries and cannibals problem?
- Connecting the Boxes [12/28/1998]
I have an arrangement of boxes and am trying to draw one continuous line
connecting them all. Can this be done?
- Constructing a Conditional Table [04/14/1998]
I have a table that is 5 columns wide and 7 rows high and contains either
a 0 or a 1 in each cell...
- Counterfeit Coin Challenge [05/12/2007]
In a set of 13 coins, either zero or two of them are counterfeit and
are lighter. You must identify the counterfeit coins, if any, after
four or fewer weighings on a balance scale.
- Counting Even Digits in Three-Digit Numbers [10/24/2004]
How many 3-digit numbers are there in which the number of even digits
- Counting Rectangles [05/23/2001]
How can I find the number of different rectangles in a square grid
containing "c" columns and "r" rows?
- Counting Rectangles Cut By a Diagonal [06/15/1999]
How can we find an equation for the number of unit squares that are cut
by a line going from corner to corner on a rectangle?
- Counting Triangles [05/27/1999]
In a large triangle with 36 small ones inside, how many triangles are
there in all?
- Covering a Checkerboard after Removing a Random Square [05/11/2008]
Use mathematical induction to prove that for any positive integer n,
if any one square is removed from a 2^n x 2^n checkerboard, then the
remaining squares can be completely (and exactly) covered with
L-shaped pieces composed of three squares.
- 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 Desert with 45 Watermelons [07/08/1998]
A boy carries 45 watermelons across a 15 mile desert, 15 at a time,
eating 1 per mile. What is the most he can carry to the other side?
- Crossing the Bridge [05/12/1997]
Four men want to cross a bridge but only two may cross at a time...
- 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?
- 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 a Cylinder out of a Sphere [02/25/1999]
What is the remaining volume after a cylinder of length 6" has been cut
through the centre of a sphere?
- Dartboard Scoring [03/01/1999]
Find the highest score you cannot get with the center worth 9 points and
the outer ring worth 4.
- Dates that Read the Same Backwards and Forwards [02/02/2010]
A student sees a palindrome in the date 01 02 2010, and wonders how to
generate all such palindromic dates. Building on another math doctor's
work with date arithmetic, Doctor Carter shares a program written in C,
then goes on to explain the purpose of each line of code.
- DEFABC = 6(ABCDEF) [12/10/2001]
Let abcdef be a 6-digit integer such that defabc is 6 times the value of
abcdef. Find the value of a + b + c + d + e + f.
- Difference of Two Cubes [05/24/2001]
The difference of two cubes is 56,765. What two positive integers satisfy
- Digital Clock Lights [06/29/1998]
If the only light source in a room is a digital alarm clock (red LEDs),
at what time is the room the darkest? Lightest?
- 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?