Ask Dr. Math

Middle School Archive

Browse Middle School Puzzles
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to frequently posed puzzles:
    Letter+number puzzles.
    Number sentences.
    Remainder/divisibility puzzles.

    1000 lockers.
    Getting across the river.
    Heads, legs: how many animals?
    A hen and a half...
    How many handshakes?
    Last one at the table.
    Measuring with two containers.
    Monkeys dividing coconuts.
    Squares in a checkerboard.
    Weighing a counterfeit coin.
    What color is my hat?

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Find Two Numbers [11/14/2001]

The sum of two numbers is 16 and their product is 48.

First 500 Even/Odd Numbers [03/14/2002]

Is there a formula for the answer to the difference of the sum of the first 500 even numbers and the sum of the first 500 odd numbers?

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?

Five Dollars in Change [10/07/1997]

I need to make $5.00. I have to use 100 coins but I can't use nickels.

Five Lines Through Twelve Dots [11/07/2000]

I have a picture made of 12 dots in a 3x4 pattern. Without lifting my pencil, how can I draw five straight lines that pass through them all?

Forming a Regular Pentagon by Folding Paper [3/11/1996]

Is it possible to form a regular pentagon from a strip of paper of any size by just folding?

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 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 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?

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.

Four Four's, 2-10 [11/24/1997]

Find a numerical expression using four 4s for each integer from 2 to 10.

Four Twos Puzzle [05/31/1999]

How can I get 15, 17, 19 and 21 using 4 twos?

Friday the 13ths in a Year [08/10/1999]

How can I figure out the maximum number of Friday the 13ths there are in one year?

Frogs Changing Sides [09/04/2002]

What is the least number of jumps needed for all the frogs to trade sides?

Fun Addition [8/21/1995]

Given the numbers 1 through 9, using each number only once, how many problems can be formed when adding two three-digit numbers? Is the sum of the digits in the sum always 18?

A Game in Three Rounds [04/19/1999]

After three rounds each of the girls has 8 chips. How many chips did they have at the start?

Game of 24 [7/15/1996]

A student takes four numbers and combines them by addition, subtraction, multiplication, division, or a combination thereof to produce an answer of 24.

Getting All the Utilities to Each House [5/17/1995]

There are three houses and three different utilities. Can you connect each utility to each home WITHOUT crossing any lines?

The Gift of the Magi [09/08/2002]

What mathematical error can be found in the first paragraph of O. Henry's 'The Gift of the Magi'?

Girls and Coins [01/29/2001]

Five girls stopped for a soda on the way home from school. The girls were amazed to find that each of them had exactly six coins...

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...

Guess and Check - How Many Coins? [10/03/2002]

Jeremy has seven fewer coins than Hanna. If together they have 83 coins, how many coins does Hanna have?

Half of Five is Four? [11/04/2004]

What's the only number system where half of five is four?

Heads and Legs [07/25/2001]

Joe counts 48 heads and 134 legs among the chickens and dogs on his farm. How many dogs and how many chickens does he have?

A Hen and a Half [09/23/1998]

If a hen and a half lays an egg and a half in a day and a half, how many eggs will five hens lay in 6 days?

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 visible faces?

Hit-and-Run Prime Number Problem [12/10/2002]

Did you get his license number?

How Does the Crystal Ball Know What Number I Chose? [01/05/2010]

Doctor Ian reveals how a mind-reading crystal ball "knows" the number you've chosen: represent any two-digit number 'ab' as 10*a + b....

How Does the Fido Puzzle Work? [05/12/2004]

Can you give me the mathematical explanation of how the Fido puzzle works? You can find the puzzle at http://digicc.com/fido/.

How Heavy is the Brick? [11/15/1997]

If a brick weighs a kilogram plus half a brick, how heavy is the brick?

How Many Games Were Tied? [06/07/1999]

How many games are tied if each team receives 3 points for a win, 1 point for a tie, and no points for a loss, and after 4 rounds the accumulated points are: Lions 22, Tigers 19, Mounties 14, Royals 12?

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 Much Money? [08/16/2001]

A man has 3 daughters, 4 sons, and a sum of money. If he divides the money equally among his daughters $2 will be left over...

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?

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