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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Multiplying by 11 [09/16/1997]

We said: 568 x 11 and after about 5 seconds she said the answer was 6248. How did she do it so quickly?

Multiply Two Numbers (No Zeros) to make 5 Billion [01/05/2002]

What two numbers, neither of them containing zeros, can be multiplied together to make 5,000,000,000?

Multi-Step Patterns [09/04/2003]

Find the rule that will give the second number if you know the first...

Nine Digits, Perfect Squares -- and Better Strategies [03/07/2010]

A puzzle about square numbers and the digits {1, 2, 3, ... , 9} leads to an insight about problem-solving approaches in general.

Number and Its Square Using All 9 Digits Exactly Once [05/22/1998]

Using multiplication facts to find all the whole numbers for which the number and its square together use exactly nine digits 1, 2, 3, ..., 9 only once.

Number Problem to Find Maximum Possible Product [02/01/2005]

Use each digit from 1 to 9 once to form two numbers such that their product is maximized. For example, the numbers could be 1234 and 56789.

Number Puzzle: 26, 70 [09/12/2001]

The sum of two numbers is 26; 4 times the first number plus 2 times the second number equals 70. What are the two numbers?

Number Puzzle Based on Multiplication Ideas [08/07/2005]

Find the smallest possible natural number that ends in 6 such that if the 6 is erased and placed in front of the remaining digits, the resulting number is 4 times as large as the original number.

Number Puzzle with Digits 1-9 in a 3 by 3 Grid [10/09/2006]

In a 3 cell by 3 cell grid, use the digits 1 through 9 to fill in the cells so each horizontal, vertical, and diagonal row of three has a sum of 15. I've tried many possibilities with no luck!

Number Riddle [1/23/1996]

What am I?: 1 is 10 cents, 14 is 20 cents, 123 is 30 cents, 943 is 30 cents

Numbers from Numbers [04/26/1997]

How do you get the number 87 only using the number 1994?

Numbers on a Star Brain Teaser [01/03/2008]

Place the numbers 20, 21, 22, 23, 24, 25, 27, 28, 29 and 31 on a star so that each line of four numbers has a sum of 100. Each number can only be used once. Is there a way to do this besides guessing and checking?

Number Strings and Reversing Numbers [02/20/2004]

I need to find an 18-digit number in which no two consecutive digits are alike and the number is such that it reverses itself upon being multiplied by 4.

Numbers with 5 Factors [02/06/2002]

I know that 16 has 5 factors (1,2,4,8,16). What other two numbers might have 5 factors?

One Billion as Product of Two Numbers with No Zeros [12/12/2001]

Write 1,000,000,000 as the product of two numbers, neither of which contains any zeros.

One Chicken, One Day [04/19/2002]

If a chicken and a half lays an egg and a half in a day and a half, how many eggs can one chicken lay in one day?

On or Off? [05/26/2001]

How can I determine whether an automatic sprinkler system is on or off at 3:30 A.M., based on timed observations by the night watchman?

The Orbit of Mercury [9/6/1995]

The number of days it takes Mercury to revolve around the sun is a palindrome. Each digit is a cube greater than one. How many days does it take Mercury to complete its orbit?

Order of Operations Problems - a General Strategy [10/23/2002]

Make equations using 7, 26, 46, and 15 to equal 160; 18, 9, 24, and 20 to equal 18.

Painting Half a Room Each Day [03/20/2002]

If Bob paints half of a room one day and decides to keep painting half each day, why is the answer infinite?

Palindromes [02/24/1998]

Two 4-digit palindromes are added to give a 5-digit palindrome N. What are the possible numbers for N?

Paper-Cutting Trick [01/24/2001]

Is there a way to cut a sheet of paper so as to make a hole big enough to walk through?

Pascal's Triangle Game [03/23/2001]

How could I make a game using Pascal's triangle?

Passing Ships [05/12/1999]

Every day a ship leaves San Francisco for Tokyo... How many Tokyo ships will each San Francisco ship meet?

Pentagon Puzzle [9/12/1996]

Three numbers are placed on each side of a five sided pentagon with the numbers on the corners being used on two sides...

Pentagon Puzzle [09/18/2002]

Write the numbers 1 through 10 so each side of the pentagon has a sum of 14.

Perilous Ping-Pong [11/14/1997]

Weigh the balls to find the one that's different...

The Perils of Predicting Patterns [11/02/2003]

What comes next in 1,3,6,10 ___? Is it 12 or 15?

The Pharaoh's Will [05/16/2000]

As he lay dying, the Pharaoh proclaimed: "I bequeath 1/3 of my estate to my oldest child; 1/4 of my estate to the next oldest child; and to each succeeding child, except the youngest, the next unit fraction of my estate; and to the youngest the remainder."

Placing Coins in a 5x5 Matrix [04/25/2000]

Each of 5 rows and colums in a grid row and column should equal a specific dollar amount: we must place 5 half-dollars, 5 quarters, 5 dimes, 5 nickels and 5 pennies in the grid...

Placing Coins That Touch [8/7/1996]

How many 20-cent coins can you put around a 20-cent coin so that all of them touch?

Planting Trees [08/13/2002]

I have to plant 10 trees in 5 rows with 4 trees in each row.

The Predetermined Sum Puzzle [02/02/2001]

I pick 5 digits, and write them down. My friend tells me a sum. Then I pick 5 more digits, he picks 5, I pick 5, and he picks 5. The sum he told me is the sum of all 5 lines. How did he know what it would be?

Price of House, Commission [8/29/1996]

If a real estate agent recieves 6 percent of the selling price of each house she sells, then what should be the sale price of a house if a seller wants to keep $61,000?

Prices in Store 88 [07/05/1999]

In Store 88 they sell exactly ten items, some items with the same price as others, but the only the digit on the price tags of each item is 8...

Prime Number Puzzle [09/27/2001]

I am the product of four primes, and the sum of these numbers is thirty. My three digits are prime and different. What number am I?

Product and Sum of Digits = Number [10/24/2001]

How many two-digit numbers exist such that when the products of their digits are added to the sums of their digits, the result is equal to the original two-digit number?

Product of Numbers 1-100 [02/23/2002]

I was wondering how to find out how many zeros will be at the end of the product of all the numbers from 1 to 100 without multiplying them all out.

Product of Terms of a Sequence [04/18/2003]

Find the product of the first 99 terms of the sequence 1/2, 2/3, 3/4, 4/5, ...

Pupil 47 Opposite Pupil 16 in the Circle [09/17/2002]

If, in Mr. Simmons' class, pupil 47 is opposite pupil 16 when the group is seated in a circle, how many students are in the class?

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