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?
- Apple Tree Line Puzzle [11/07/1997]
Plant 13 apple trees in 9 rows with 4 apple trees in each row.
- Arrange 7 Points in a Plane... [10/05/1998]
Arrange 7 points in a plane so that if any three are chosen, at least 2
of them will be a unit distance apart.
- Calendar Trick [12/04/1996]
Given a day, month, and year, is there a formula you can use to find the
day of the week?
- Diagonals and Tiles [11/17/2001]
Jay tiled a 15x21' rectangular ballroom with 1 ft. sq. tiles. Then he
drew diagonals connecting opposite corners of the room. How many tiles
did the diagonals pass through?
- Dimensional Analysis [10/06/2001]
Mr. R wants to give all of his chemistry students enough chocolate to
make them goofy for the rest of the day. It's a known fact that it takes
47 chocolate chips to make a student goofy...
- Finding One Coin of 12 in 3 Steps [8/6/1996]
Given a pile of twelve coins of equal size with one of a different
weight, in three weighings find the unequal coin...
- Find the Four Integers [06/10/2003]
Four distinct integers a, b, c, and d, have the property that when
added in pairs the sums 16,19,20,21,22, and 25 are obtained.
- Find the Seven Numbers [06/07/2003]
There are seven numbers, A, B, C, D, E, F, and G. When all these
numbers except A are added together, 116 is obtained...
- Formula for Nim [02/22/2002]
Is there a formula for the game of Nim?
- The Four Fours - an Ancient Problem [10/02/1997]
The problem allows you to use only four fours and as many operands as you
like in order to create mathematical sums that equal the numbers from 1
to as high as you can go.
- Grid with 14 Dots [11/10/1997]
Put dots in 14 squares in a 6 by 6 grid. There must be an even number of
dots in every row and column (zero is even). Is there a rule for building
solutions to this problem?
- Hands of a Clock [10/10/1997]
How many times do the hour and minute hands cross in a 12-hour period of
- How Many Cube Faces Were Painted? [02/23/2003]
Some unit cubes are put together to form a larger cube. Some of the
larger cube's faces are painted and then it is taken apart. 45 small
cubes are found to have no paint on them. How many faces of the large
cube were painted?
- Letter and Word Puzzle: Bag = 2 + 1 + 7 = 10 [02/20/2003]
What words can I use to make 10?
- Look-and-Say Sequence [02/14/2002]
I can't find the next six numbers: 1, 11, 21, 1211, 111221, ...
- Lucky Seven Fractions Puzzle [12/22/2001]
Put numbers 1-9 in order to make the equation correct: XX/XXX+XX/XX=7.
- Magic Pentagon [12/02/2001]
Using the numbers 1-10 once each, make the sums of the three circles
- Mensa: Numbering for an Alternate World [5/25/1995]
In a parallel universe, the numbering system in use is based on the 26-
character Roman alphabet. In this universe you are driving from New York
to San Francisco. A road sign indicates you are BBQ miles from San
Francisco. Are you closer to...
- Number of Squares in an NxN Square [7/29/1996]
How many squares are there in an 8 x 8 square? How many rectangles are
- Order of Operations Puzzle [09/22/2002]
What is _ + _ x _ - _ = 22? We're given the numbers 2, 3, 4, 8.
- Pascal's Triangle and the Tower of Hanoi [08/07/1997]
There's a connection between Pascal's Triangle and the Tower of Hanoi
game but I can't remember what it is!
- Proving that 1 + 1 = 3 [11/18/1995]
How can I show that 1 + 1 = 3 ?
- Remainder Problem [10/27/2001]
What number less than 500 produces remainder 4 when divided by 5,
remainder 7 when divided by 9, and remainder 9 when divided by 11?
- Remainders of 1, 2, 3, 4 [10/09/2001]
Find the smallest whole number that when divided by 5, 7, 9, and 11 gives
remainders of 1, 2, 3, and 4 respectively.
- Solving Number Sentences: A Strategy [07/11/2003]
I have been given a list of numbers without any plus or minus signs
and I have to place the signs between the digits.
- The Traveling Bee [09/18/1998]
If a bee travels between two trains that are moving at 30 and 20 mi/hr
respectively, starting from 50 mi apart, how far does the bee travel?
- Trying to Solve a Challenging Number Puzzle [07/28/2006]
Working an a number puzzle that seems impossible? Think you must have
to do some form of lateral or unusual thinking? Read this first!
- Ways To List 1, 2, ... 10 Out of Order [02/25/1997]
How many ways are there to list the numbers one through ten so that no
number appears in its own position (i.e. 1 is not first in the list, 2 is
- 1000 $1 Bills in 10 Envelopes [12/11/2002]
Put 1000 $1 bills into 10 envelopes in such a way that someone can ask
you for any amount of money from $1 to $1000 and you can give it to
him through a combination of the envelopes.
- 1000 Lockers [11/06/1997]
The 1st student opens all 1000 lockers, the 2nd student closes lockers
2,4,6,8,10, etc., the 3rd student opens lockers closed and closes lockers
open on lockers 3,6,9,12,15...
- 100 Lockers, 100 Students [8/16/1996]
There are 100 closed lockers in a hallway, and 100 students. Student 1
walks down the hall, opening every locker...
- 10-Digit Number Puzzle [02/10/2001]
How many possibilities are there for the 10-digit number abcdefghij if
all of its digits are different and it is evenly divisible by 11111?
- 10 Digit Number Puzzle [09/14/2004]
Write a 10 digit number where the first digit tells you the number of
zeros in the 10 digit number, the second digit is the number of ones
in the number, the third digit is the number of twos in the number and
so on until the 10th digit is the number of nines in the number.
- 10 Square Units on Dot Paper [03/01/2003]
How can I draw a square with 10 square units on dot paper?
- 121, 111211, 311221 Puzzle (Look and Say Sequence) [10/23/2001]
121, 111211, 311221 - what's the next number?
- 12345679 Puzzle - The Missing Eight [08/17/2001]
Start with 12345679 and pick a number...
- 16 Squares and 5 Colors [08/15/2002]
You have 16 squares in a box, with 5 colors: 4 blue, 3 green, 3 red, 3
white, and 3 black. The same colors can't touch or be in the same row.
- 1, 7, 23, 55, 109, 191, ___ [10/03/2002]
My family is stumped on this number pattern: 1,7,23,55,109,191,___ ...
- 174 Game [08/16/2001]
Players announce numbers (1-8) in turn. The judge adds the numbers they
announce. If the sum becomes 174 or more, that player wins. In order to
win the game, which number should be given first?
- 17 and 19 Balls, Colliding [03/25/2003]
How many collisions will occur in the described system?