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?
- Geometry Puzzles [12/18/1995]
A student asks Dr. Math for help in finding the correct combinations of
numbers to solve two puzzles.
- Giving Myself a Challenge [09/20/2014]
A student wonders if doing puzzles from math competitions would help her as she prepares for her advanced coursework. Doctor Ian suggests many ways to exercise lateral thinking skills, mathematical insight, and creativity in problem-solving no matter the source of a question &mdash or its difficulty.
- Grains of Wheat [07/14/1999]
The person who invented the game of chess was said to have been offered
any payment he wanted... How much wheat did he receive?
- Grandfather Clock and 7-Second Chime [06/06/2002]
If it takes a grandfather clock 7 seconds to chime 7 o'clock, how long
will it take the same clock to chime 10 o'clock?
- Grid Game [07/20/2001]
This game for two players is played on a rectangular grid with a fixed
number of rows and columns. Play begins in the bottom left-hand square...
- Guessing a Mystery Number [05/23/2007]
One person thinks of a natural number and the other person has to try
to guess it by asking questions. The one who knows the number will lie
once at some point, but otherwise be truthful in answering the
questions. What's the best algorithm for finding the number, and
what's the fewest guesses it would take?
- Half of Five is Four? [11/04/2004]
What's the only number system where half of five is four?
- Handshake Problem Variant [07/08/1999]
Five couples go to a party and start shaking hands. One of the men
shouts, "Stop! How many hands did you shake?" Every person gives a
- The Haybaler Problem: Reflections and Refinements [10/05/2002]
Given pairwise weights, a student and Doctor Greenie reflect on how to deduce
individual ones before a PhD follows up with several simplifying insights of his own.
- Headscratch Letter Puzzle [01/25/2002]
Use the numbers 0 to 9 for each letter once: G*B=HA, D*E=E, I*C=C, H*
- Hidden Faces in a Set of Cubes [10/04/2000]
Can you give us a hint for a formula that will tell you the number of
hidden faces in an arrangement of a cubes if you know the number of
- Hole in a Sphere [12/30/1996]
When you bore a 6 inch cylindrical hole through the center of a sphere,
what is the volume of the remaining solid?
- How Many Friday the 13ths [9/12/1996]
Is there a way to figure how many Friday the 13ths can occur in a given
- How Many Handshakes? [01/29/2001]
There are 40 people in a room. They shake each other's hands once and
only once. How many handshakes are there altogether?
- How Many Hidden Faces? [07/07/2002]
When you place a number of cubes in a row on a surface, how many of the
faces can't be seen from any position? Is there a formula for this?
- How Many Pencils? [09/05/2001]
How many pencils does Al have if all of them are blue except 2, all of
them are yellow except 2, and all of them are red except 2?
- How Many People Went on the Cruise? [12/03/2001]
At the end of a special cruise, the employees could not remember the
total number of people who were on board. However, they had the following
data from the passenger list: 520 European females...
- How Old is Korinth? [09/09/1997]
Korinth is twice as old as Marin was when Korinth was as old as Marin is
now. Marin is 18.
- How to Create a 4 x 4 Magic Square [01/28/2004]
Is there a general technique for creating a 4 x 4 magic square so that
the rows, columns, diagonals, four center squares and four outside
corner squares each add up to a desired number?
- The Hundred Fowls [09/29/2001]
If a rooster is worth five coins, a hen three coins, and three chickens
together are worth one coin, how many roosters, hens, and chickens
totaling 100 can be bought for 100 coins?
- If 1/x+x=5, what does 1/x^2+x^2 equal? [08/16/2002]
A complex and an easy solution to the problem.
- Indefinite Series, Perfect Squares [01/01/1998]
Across the first row of an 11-column table are the numbers 1991, 1992,
1993, .... 2000, 2001.
- Jack is Older than Jill [11/16/2001]
If you reverse the digits in Jack's age, you get Jill's age. The sum of
their ages equals 11 times the difference between them. Jack is older
than Jill. What are their ages?
- Knights of the Round Table [07/01/1998]
If x knights are sitting at a round table, and every other one is
removed, who is the last one left sitting at the table?
- Largest 7-Digit Number [07/27/2001]
Work out the largest 7-digit number you can applying two rules: every
digit in the number must be able to be divided into the number, and no
digit can be repeated.
- Largest Magic Square Ever Known [09/18/2001]
What is the largest magic square ever constructed?
- Last Digit of a Number [04/14/1997]
What is the last digit in (1996^1997)-(1997^1996)?
- Laying Eggs Better by Half [12/05/2001]
If a hen and a half lays an egg and a half in a day and a half, how many
and a half that lay better by half will lay half a score and a half in a
week and a half?
- Least Perimeter [10/13/2002]
Finding a formula for least perimeter of a square or rectangle.
- Left-Sided Rhombuses in a Larger Rhombus [05/22/2000]
How many left-sided, right-sided, and vertical rhombuses can be found in
a larger NxN rhombus?
- Let's Make a Deal Probability [06/01/1999]
In "Let's Make a Deal," does Monty's showing you a door, or his knowing
which door the prize is behind, affect the probability?
- Linear Equations and Functions [10/11/1997]
The digits of a positive 2-digit integer N are interchanged to form an
integer K. Find all possibilities for N... the average of N, K, and 35 is
- Line Drawn through Lines Puzzle [10/18/2001]
Given a box made up of 16 lines, with two rectangles above and three
squares below, draw a line through each line without crossing any line
- Links from 1 to 63 [02/19/2002]
You are given 1 chain of 63 links to use instead of money. Your task
is to cut the chain in 3 places so that you'll be able to hand a
person any number of links he asks for, from 1-63.
- Locker Problem [11/21/1997]
There are 1,000 lockers numbered from 1 through 1,000. The first student
opens all the doors; the second student closes all the doors with even
- Logic - Liars & Truthtellers (What Question Does She Ask?) [3/12/1995]
A logician vacationing in the South Seas finds herself on an island
inhabited by the two proverbial tribes of liars and truth-tellers.
- Longest Roman Numeral [05/03/2007]
Which year in our recorded history has had the greatest number of
Roman numerals in it?
- Lucky Seven Fractions Problem [06/08/2002]
Substitute each of the digits 1-9 for one '?' in ??/??? + ??/?? = 7 to
make the equation true.
- Magic Square [5/16/1996]
How can I make a magic square?
- Magic Squares [10/19/2003]
A lengthy conversation about magic squares that includes a number of
interesting patterns and ways to create them.