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?
- Disproving that One is Equal to Two [11/13/1995]
I'm trying to disprove a proof that says 1 is equal to two. I've been
struggling with finding the error in my logic for quite some time now,
but can't seem to quite get where I messed up.
- Distance between Points on a Line [10/02/2002]
When the 10 distances between 5 pairs of points on a line are listed
from smallest to largest, the list reads: 2,4,5,7,8,k,13,15,17,19.
What is the value of k?
- Dividing the Prize Money [05/24/2000]
Mrs. Shoe gave her prize winnings to her children in order... and all the
prize money was divided equally amongst her children. How many children
- Divisibility of a 180-Digit Number by 99 [05/10/2007]
Consider the 180 digit number formed by putting the numbers from 10 to
99 together consecutively (1011121314...99). If you divide this number
by 99, what is the remainder?
- Divisibility Puzzle [01/22/2001]
The leftmost digit of an integer of length digits is 3. In this integer,
any two consecutive digits must be divisible by 17 or 23. The 2000th
digit may be either a or b. What is the value of a+b?
- Divisibility Word Problem [1/27/1996]
Arrange the digits 0 to 9 such that the number formed by the first digit
is divisible by 1, the number formed by the first two digits is divisible
by 2, that formed by the first three digits divisible by 3, and so forth;
thus the number formed by the first 9 digits will be divisible by 9 and
that formed by all 10 digits divisible by 10.
- Doubling Pennies [11/26/1996]
If I start with a penny and double it daily for 30 days, how many pennies
do I have at the end?
- Doubling Pennies for a Year [06/03/2001]
How many pennies will we have at the end of 365 days if we begin on
January 1 and double our money each day?
- Employee Scheduling [09/22/1998]
Can you help me make a schedule to staff an ice cream parlor?
- Enumeration in Combinatorial Analysis [11/28/1996]
How many 3 digit numbers have the digital sum of nine?
- EVE/DID = .TALKTALKTALK... [11/21/2000]
In the equation EVE/DID = .TALKTALKTALK... each letter corresponds to a
different number, and EVE/DID is in lowest terms. What does each letter
- Even-Numbered Magic Squares [04/30/2001]
How can I construct magic squares that work for even numbers?
- Exchanging Seats in a Boat [06/08/2001]
Ten women are fishing in a long, narrow boat. One seat in the center of
the boat is empty. The five women in the front of the boat want to change
seats with the five women in the back of the boat...
- Extraordinary Social Security Number [12/05/2001]
The number's nine digits contain all the digits from 1 to 9. When read
from left to right the first two digits form a number divisible by two,
the first three digits form a number divisible by three...
- Feeding Oxen [6/26/1996]
We have 3 pastures with grass of identical height, density and growth
rate... How many oxen can be fed for 18 weeks?
- A Fibonacci Jigsaw Puzzle [04/29/1999]
Why is the area of our rectangle, formed from a square, 65 when the
square's area was 64?
- Fibonacci Riddle [11/21/2001]
We can cut an 8x8 square with an area of 64 into four pieces and
reassemble to get a 5x13 rectangle with an area of 65. Where does the
extra 1x1 square come from?
- Finding a Desired Perfect Cube [04/26/2007]
What is the smallest positive cube that ends with the digits 2007?
- Finding A Number Given Its Divisors and Remainders [10/22/2003]
A general strategy for solving problems such as finding the smallest
whole number that when divided by 5, 7, 9, and 11 gives remainders of
1, 2, 3, and 4 respectively.
- Finding a Rule to Fit Given Data [11/14/2005]
Given that 1J1 = 2, 3J5 = 34, 6J9 = 117, and 10J14 = 296, conjecture a
value for 3J8.
- Finding a Sequence of Numbers [08/13/2002]
Given six digits in a sequence: two 4's, two 5's, and two 6's, with
one digit between the two 4's, two digits between the two 5's; and
three digits between the two 6's, write this sequence of numbers.
- Finding a Six-Digit Number [05/12/2005]
In a 6-digit number, the sum of the digits is 43, and only two of the
following three statements about the number are true: (1) It's a
perfect square. (2) It's a perfect cube. (3) It's less than 500,000.
- Finding Sets of 7 Prime Numbers That Sum to 100 [05/22/2007]
There are 35 different sets of 7 prime numbers that sum to 100. Of
those sets, which has the largest product, and which has the largest
number? I'm using trial and error and it's very frustrating. Is there
a better way?
- Finding the Next Number in a Series [07/22/2002]
Are there any formal or systematic methods for solving problems that
ask you to find the next number in a series?
- Find Number Given Divisor and Remainder Information [01/21/2009]
A four digit number N leaves remainder 10 when divided by 21, remainder
11 when divided by 23 and remainder 12 when divided by 25. What is the
sum of the digits of N?
- Find Pairs of Positive Integers [10/15/2001]
Find pairs of positive integers whose greatest common factor is 1 and
whose sum is 2000.
- Find the 3-Digit Values A, B and C [08/08/1999]
How can I determine the 3-digit integers a, b, and c when the ratio a:b:c
is 1:3:5 and the digits of a, b, and c are 1,2,3,...,9, each appearing
- Find the Pattern, Complete the Combinations [01/13/2002]
Find the pattern: 3*4=5, 8*4=0, 3*7=2, 1*2=9, and use it to complete the
combinations: 5*5=?, 4*4=?, 5*7=?
- The 'First to 100' Game [03/12/2001]
Two players take turns choosing any number from 1-10, keeping a running
sum of all the numbers. The first player to make this sum exactly 100 is
the winner. Is there a surefire way to win this game?
- Formula for Easter [06/12/1999]
Is there a formula for finding the month and the day on which Easter
falls in a given year?
- Formula for Factors of a Number [11/3/1996]
How many triangles can you draw on a square grid of dots of size x*x?
- Formula for Squares inside Rectangles [06/29/2003]
Is there an equation for how many squares there are in a rectangle
divided up into 1cm blocks?
- Formula for the Day of the Month [04/05/2001]
I have read the Dr. Math FAQ on finding the day of the week from the
date. How do you get the part of the equation that deals with the month?
- Formula for the Day of the Week [05/21/1997]
Please tell us what day of the week the Declaration of Independence was
signed on, and the formula to determine it.
- Four 4's Puzzle [10/24/2001]
How can you get 73 by using 4 fours and any mathematical equation?
- Four Dogs Running [08/08/2001]
Four dogs are at four corners of a field. Each dog chases the dog to its
right; all four run at the same speed and no acceleration is assumed.
Where will they meet, and how long and how far will they have run when
- The Four Doors of Xanth [02/11/1998]
Each door conceals one item: a treasure, a rope, a key, and a lantern.
You must find all four items in a particular order to keep the treasure.
- Freezer Roulette [07/26/2002]
Two people are trapped in a small freezer that is slowly getting
colder and colder... Is it possible for them to escape? If so, how?
What’s the maximum number of minutes required to escape given ANY
initial combination of button states?
- Frogs Changing Sides [09/04/2002]
What is the least number of jumps needed for all the frogs to trade
- A Geometry and a Logic Problem [3/11/1995]
Problem 1: A cylindrical hole six inches long is drilled straight through
the center of a solid sphere. What is the volume remaining in the sphere?
Problem 2: The classical stay-switch problem.