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?

---

Thinking Outside the Pen [09/28/2004]

How can you put 21 pigs in 4 pig pens and still have an odd number of pigs in each pen?

Thoughtful Trial and Error [05/21/2002]

Arrange the numbers 1 through 10 in a triangle so that all the rows of three numbers have the same sum, and all the rows of four numbers have the same sum.

Three Cuts and Seven Pieces [01/18/1999]

Is there any way that you can cut a pie into seven pieces with just three straight cuts?

Three Holes Puzzle [05/02/2002]

A piece of plywood has three holes it it: a circular hole with a diameter of 2 cm, a square hole with 2 cm sides, and a triangular hole with a base and height of 2 cm. What object could completely plug AND pass completely through each hole?

Three Jars [11/01/2000]

One jar has red marbles, one has black, and one has both, but their labels are wrong. Can you fix the labels by picking one marble from one jar?

Three Weights [12/07/1997]

A boy selling fruits has only three weights, but with them he can weigh any whole number of pounds from 1 pound to 13 pounds inclusive. What three weights does he have?

To Find the Heavy Marble, Divide and Conquer Asymmetrically [01/14/2011]

A student knows how to deduce the one heavy marble out of eight by weighing successive subsets on a scale balance three times ... but after only two weighings? Doctor Ali suggests the first step.

Toothpick Puzzle [09/27/2004]

If you have a square that is made up of nine little squares, each one toothpick per side, so the big square is three toothpicks per side, can you remove 5 toothpicks and leave 3 squares (of any size)?

Tower of Hanoi [10/16/1997]

I'm looking for a mathematical solution, not a trial-and-error one.

Towers of Hanoi Puzzle: 3 Pegs, n Discs... [11/15/1997]

We are told to find out the least number of moves it takes to get three discs in size order onto the third peg from the first.

Tracing a Figure Without Lifting Your Pencil [03/09/2001]

Is there a simple way to quickly tell whether a figure can be traced without lifting your pencil?

Trains and Fractions [12/24/1994]

A non-stop train leaves city A for city B at 60 m.p.h.... If to the numerator and denominator of the fraction 1/3 you add its denominator... A train moving at 45 m.p.h. meets and is passed by a train moving at 36 m.p.h...

The Traveling Businessman Dilemma [04/09/1998]

Three businessmen are on a trip and need a place to stay for the night...

Twenty or Fewer Steps [08/28/2002]

Only the 1, +, -, x, /, (, ), and = keys on a scientific calculator are working. How can a result of 75 be reached by pushing these keys fewer than 20 times?

Two 2s Make 5 [01/03/2003]

Using only two 2s and any of the standard mathematical symbols, write an expression whose value is equal to exactly five.

Two Mathematicians: Factoring Logic [03/24/2003]

Two mathematicians are each assigned a positive integer. They are told that the product of the two numbers is either 8 or 16. Neither knows the other's number...

Two Mathematicians Problem [05/18/1998]

One mathematician is give the sum of integers X and Y, and another is given their product... what are the numbers?

Two Numbers with Equal Sum, Product, and Quotient? [05/07/2007]

Find two numbers such that when you find the sum, product, and quotient of the two numbers they are all equal.

Two Quantities, Two Relations [04/04/2000]

Find two consecutive numbers whose squares, when subtracted, equal 43.

Unit Fractions and the Greedy Algorithm [12/27/2000]

How can I represent 2000/2001 as the sum of unit fractions?

Unit Fractions Summing to 1 [07/15/2001]

Find seven different unit fractions whose sum is 1.

Units Digit of 2^57 [10/07/2001]

What is the units digit of 2 to the power of 57?

U.S. and European Sock Sizes [03/23/2002]

Which expression could be used to convert European size to U.S. size?

Use 10 Digits in 2 Fractions that Add to 1 [05/26/2003]

Obtain a sum of the form xx/xxx + xx/xxx = 1, using all digits 0-9 exactly once.

Use 2,3,7,8 to Write Math Expressions from 1 - 50 [10/03/2004]

Using each of the digits 2,3,7,8 one time, and any math operations you know, find expressions with values of 44 and 46. I have done from 1 to 50 except these last two numbers.

Use One Weighing [02/06/2002]

There are 90 genuine coins and 10 fake coins in 10 stacks with 10 coins in each stack... Using only one weighing, determine which stack contains the fake coins.

Using Prime Factors to Limit Search [07/26/2002]

Our courtyard has more than one tree, and each tree contains more than one bird...

The Value of a Word [08/25/1998]

Think of a word that equals one dollar. The key is: a=.01, b=.02, c=.03, ....

Venn Diagram: Goops, Gorps, Gorgs [09/19/2002]

Every Goop is a Gorp. Half of all Gorgs are Gorps. Half of all Gorps are Goops. There are 40 Gorgs and 30 Goops. No Gorg is a Goop. How many Gorps are neither Goops nor Gorgs?

Visualizing Two Word Problems [03/12/1998]

Sandy has thrice as many beads as Wendy... Paul gives 20 of his marbles to Simon...

Weighing Bolts [2/14/1996]

A hardware plant makes rivets 6 inches long and 6 ounces in weight. Harry makes a drum of rivets 6 inches long weighing only 5 ounces each. Harry's drum is stored along with 9 other proper drums. Harry has to find the rivets but he can only use a scale one time, and can take only one reading from that scale. How can Harry be sure of picking the right drum out of the ten?

Weight of Each Bale of Hay [09/26/1997]

Five bales of hay are weighed in all possible combinations of two...

What's His Street Address? How Many Neighbors Does He Have? [09/29/2010]

A student struggles with a word problem that asks for specific sums of counting numbers. Three different doctors weigh in with increasingly sophisticated and comprehensive problem-solving approaches: programming spreadsheet formulas; applying combinatorics; and invoking quadratic Diophantine and Pell equations.

Where Do the Parentheses Go? [09/12/2003]

Add parentheses to 2 + 7^2 - 3^2 / 3^2 - 1 * 5 = 35 to make it a true statement.

Where to Put the Parentheses? [01/03/2002]

I am stuck on where to put parentheses in the math expression 9-6+4*6/ 3=-2 to make the expression true.

Which Car is in the Pole Position? [08/14/1999]

Clues for finding starting positions of the 10 cars entered in the Rithmetic Road Rally.

Which Grasshopper Wins the Race? [06/11/2003]

The larger grasshopper covers 10 inches per jump, while the smaller one covers 5 inches. The smaller grasshopper takes 2 jumps for every jump of the larger grasshopper. Which one wins, and why?

Who Finished In What Place? [12/15/2002]

Sarah, Carla, and Kim were the finalists at a gymnastics meet...

Who Made Which Toys? [12/21/1998]

A math logic problem, from a rhyme describing Santa's toymakers.

Why the Motionless Runner Parodox Fails [01/01/2005]

I read about the Motionless Runner paradox on your site, and I am now convinced that motion is an illusion. Can you help me understand why the paradox can't be true?

Page: [<prev]  1  2  3  4  5  6  7  8  9 10 [next>]