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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Puzzle with a Difference [03/19/2003]

Place each number from 1 through 10 in a box. Each box must contain a number that is the difference of two boxes above it, if there are two above it.

Pyramid Problem [9/1/1996]

You have a pyramid (1 circle on the top layer, 2 on the second, 3 on the third, 4 on the fourth) and you can only move three circles to turn it upside down...

Quickly Finding the Day of the Week [11/14/2000]

Today is November 14, 2000, a Tuesday. What day of the week was November 14, 1901?

Rational Number and its Reciprocal [03/14/2002]

A rational number greater than one and its reciprocal have a sum of 2 1/6. What is this number? Express your answer as an improper fraction in lowest terms.

Rectangles on a Chessboard [02/09/2002]

How many rectangles are there on a chessboard?

Reversing the Digits [05/29/2003]

Finding pairs of two-digit numbers that yield the same product when you reverse their digits.

Russian Nim [02/15/1999]

Strategies for winning at Russian Nim (the "20" game).

Send More Money [05/12/1997]

Find the digit that each letter represents in the equation SEND + MORE = MONEY.

Sequence Question from IQ Test [06/28/2007]

Find the missing number in the sequence 11 > ? > 1045 > 10445.

Simultaneous Equations with Integral Solutions [11/29/1996]

What kind of a math project could I do with magic squares?

Six Lines, 4 Triangles [8/19/1996]

How can you form four triangles from six toothpicks?

Six Lines Make Twelve Triangles [11/06/2002]

How can I use 6 lines to make 12 triangles?

Skilled and Semi-Skilled Workers [09/04/2002]

Four skilled workers do a job in 5 days, and five semi-skilled workers do the same job in 6 days. How many days will it take for two skilled and one semi-skilled worker to do that job?

Solving a 3 x 3 Magic Square [09/29/2005]

There are mechanical methods to fill in Magic Squares, but here Dr. Wilko presents a nice way to reason out the solution of a 3 by 3 square.

Solving a Math Poem [05/24/2000]

Take five times which plus half of what, and make the square of what you've got...

Solving Problems by Making Organized Lists [04/09/2008]

To find 11 coins that total $1.37, Dr. Ian makes organized lists which reduce the problem to smaller and smaller problems until it can be solved. This general strategy is useful in many math problems.

Solving Questions [08/28/2002]

In a poll of 34 students, 16 felt confident solving quantitative comparison questions, 20 felt confident solving multiple choice questions.... How many students felt confident solving only multiple choice questions and no others?

Solving SEND + MORE = MONEY [04/18/2002]

I have tried logical reasoning and can't get it.

A Special Ten-digit Number [02/17/1999]

Create a ten-digit number that meets some special conditions...

Splitting a Clock Face into Desired Sums [11/14/2007]

Break a clock into exactly five pieces such that the sums of all the numbers on each piece are 8, 10, 12, 14 and 16.

Squares in a Square [01/23/2003]

If you have a 50x50 square with small squares inside it, how many squares will there be altogether?

Squares in Rectangle Formula [06/30/2003]

What is the equation for the number of squares in a rectangle (like the chessboard puzzle)?

Squares on a Checkerboard [04/26/1998]

How many squares are there on a checkerboard?

Squares, Rectangles on a Chessboard [08/14/1997]

How many squares are there on a chessboard? How many rectangles?

Squaring Two-Digit Numbers Ending in 5 [09/10/2001]

Take the first digit, multiply it by the next consecutive number, and place it in front of 25. Can you prove this shortcut?

Stair Patterns [02/27/2001]

The 1st step is made with 4 matches, the 2nd with 10 matches, the 3rd with 18, the fourth with 28. How many matches would be needed to build 6, 10, and 50 steps?

Subtraction Pattern for Roman Numerals [9/7/1995]

The question is, given that 4 is 'IV' and 9 is 'IX' and 900 is 'CM', does the subtraction pattern follow for two numerals more than two 'levels' apart, and can numerals which represent numbers starting with 5 be subtracted? For example, would 99 be 'IC', would 450 be 'LD', and would 995 be 'VM'?

Sum of Numbers 1-500 [06/20/2001]

What is the formula to find the sum of the numbers one to five hundred?

Sums Divisible by 11 [10/10/2001]

Why is the sum of a number with an even number of digits and that same number written in reverse always divisible by 11?

Sums of Consecutive Integers with Digital Sums [01/27/2004]

Find all sets of positive consecutive integers that sum to 100, and whose digits sum to greater than 30.

Sums of Sets of Prime Numbers [01/07/2003]

Given several sets of prime numbers, use each of the nine non-zero digits exactly once. What is the smallest possible sum such a set could have?

Sum Twice the Difference [10/18/2002]

What number can you add to and subtract from 129 such that the sum is twice the difference?

Swimming Laps [04/15/2002]

John decides to swim a certain number of laps of the pool in five days. On the first day he covers one fifth of the total. The next day he swims one third of the remaining laps...

Switching Dollars and Cents [10/07/1997]

How do I find an equation?

Tea and Cakes [11/13/2002]

A cafe sold tea at 30 cents a cup and cakes at 50 cents each. Everyone in a group had the same number of cups of tea and the same number of cakes. The bill came to $13.30. How many cups of tea did each person have?

Teaching Elementary Probability [10/07/1998]

If you toss a number cube 20 times, could it land on six 20 times?

Ten Card Arrangement Puzzle [10/04/2005]

Arrange ten cards numbered 1-10 in a pile. Turn over the top card, then move the next card to the bottom of the pile. Turn over the new top card and move the next card to the bottom of the pile. Continue like this until all ten cards have been turned over. The challenge is to arrange the pile so the cards are turned over in order from 1 to 10.

Terms and Rules [12/14/2001]

My daughter is in sixth grade and has been doing a pattern journal where she has two columns of numbers; the first column is the n and the second column is the term, and she has to find the rule.

Thinking about Number Pattern Problems [08/09/2004]

How do I figure out the next 2 numbers in the pattern 1, 8, 27, 64, ____, ____?

Thinking about the Maximum Area Enclosed by a Fence [04/15/2004]

You have 2000 meters of fencing. What is the largest area you can enclose with it using various shapes?

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