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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1 Dollar, 50 Coins [02/22/2002]

I'm looking for two ways to change a dollar bill using exactly 50 coins...

50 Coins = One Dollar [02/24/1997]

How can you make up one dollar using 50 coins?

5 Coins = 55 Cents [01/28/1997]

I have five coins in my pocket. Together they are worth 55 cents. What coins do I have?

Addition and Multiplication with Roman Numerals [8/27/1995]

How did Romans do Math with Roman Numerals?

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?

Change for a Dollar [01/22/2001]

Could you please tell me the two hundred plus ways in which coins are used to make change for one dollar?

Coin Combinations to Make a Dollar [11/12/1996]

What is the number of combinations you need to get a dollar using pennies, nickles, dimes, quarters, and half dollars?

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...

Farmer Crossing a River [09/24/2001]

A farmer is taking his prize lettuce, lion, llama, and pet leviathan (a unique creature that eats llamas and lions unless a lettuce is present) to market...

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...

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 time?

The Hands on a Clock [2/1/1996]

How many times in a 24 hour period do the hour and minute hands meet?

How Does the Trick Work? [06/20/2003]

From time to time I've run across math tricks that ask me to give a number, then take me through a series of other questions and by the time I work my way to the end, the trick tells me my original number.

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?

How Many People? [06/11/2001]

There are three more people in front of Clancy than in front of Nancy, and twice as many in front of Mary Ellen as in front of Nancy. The total people in front of the girls is 11. How many people are in front of each of the girls?

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 equal.

The Motel Bellboy and the Missing Dollar [01/28/1998]

The bellboy decides to give each of the men only one dollar back and keep the other two dollars for himself...

Multiples of 9 Number Puzzle [03/23/2003]

If you take a two-digit number, add the digits, and subtract the sum from the original number, why is the answer always a multiple of 9?

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 there?

Number Puzzle Solved with and without Algebra [12/10/2001]

I am a number. Add 25 to me. Divide the sum by 2. You end with 75. What number am I?

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!

Ratios of Ripped Paper [11/17/1995]

Rip a piece of 8 by 11 paper in half, then put that half over the other, then rip it again. What is the length (or how high, like 5 miles or something) when the paper is ripped 30 times?

Rectangles and Factors [09/09/1998]

How many rectangles can you make with 10 small squares? 5 small squares? 12 small squares?

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.

Squaring Pennies [9/20/1995]

Is it better to have $1,000,000 or 1 penny doubled every day for a month?

Sum of numbers and/or digits 1-100 [03/29/1997]

How would one go about adding up all the digits 1-100?

Sum of Odd Numbers 1-50? [10/10/2001]

What is the sum of the first 50 odd numbers?

Tricky Word Problem - The Missing Dollar [2/26/1996]

A student asks for help with what seems to be an impossible word problem (more on 'Dividing the Cost of a Hotel Room', above).

Twelve Days of Christmas Gifts [12/06/2001]

How many presents did the person receive in the song "The 12 Days of Christmas"?

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 not second...)?

1=0? [11/8/1994]

A group of students in an eighth grade math class here want to know how to prove that one actually equals zero. Is it something like using the quadratic formula?

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