Ask Dr. Math

High School Archive

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

Selected answers to frequently posed puzzles:
    1000 lockers.
    Letter+number puzzles.
    Getting across the river.
    How many handshakes?
    Last one at the table.
    Monkeys dividing coconuts.
    Remainder/divisibility puzzles.
    Squares in a checkerboard.
    Weighing a counterfeit coin.
    What color is my hat?

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19 Apple Trees [06/29/2001]

You have 19 apple trees. How could you place them in 9 rows of 5?

The 22 Puzzle [03/12/2002]

Choose three different digits from 1-9, make all the 2-digit numbers you can from these 3 digits... why is the answer always 22?

2^4 = 16 AND 4^2 = 16 [10/29/2001]

Can you think of any other pair of unequal numbers that share the same relation as 2 and 4 in the above example? What was your strategy?

300 Foul Shots [07/11/2001]

A basketball player shoots 100 foul shots on a given day, and increases the number he shoots by 10 per day...

30th of the Month [06/18/2001]

What day does the 30th of the month most often fall on?

33 Pearls Problem [11/14/1996]

On a string of 33 pearls, the middle pearl is the most expensive one - find its value.

36 Sums, Blank Dice [03/15/2001]

You have two blank, six-sided dice, and you can put any numbers on them. The 12 numbers you choose should allow you to make the sums from 1-36...

3 Digits Plus 3 Digits [12/09/2001]

Use the digits 1 to 9 only once in a sum that must be a three-digit number plus another three-digit number to equal another three-digit number. Each digit can only be used once, but all must be used.

3-D Shape That Can Be a Circle, Square, or Triangle in 2-D [09/08/2006]

Is there a shape that can fill and pass through a circular hole, a square hole, and a triangular hole?

3 Weighings Problem [12/09/1997]

You have a balance scale and 12 balls that all look and weigh the same except one...

4(ABCD) = DCBA [06/15/1997]

Find a four digit number such that 4(ABCD) = DCBA.

9 x HATBOX = 4 x BOXHAT [05/30/2000]

What is the solution to the cryptogram: 9 x HATBOX = 4 x BOXHAT?

ABCDC - BEAAC = BADAD [03/25/2002]

What do the letters stand for if D = 0?

ABCDE/4 = EDCBA [06/30/2003]

Find a five-digit number that, when it is quartered, gives an answer which is its digits in reverse order.

A,B,C,D Letter Puzzle [08/30/2002]

Place the letters A,B,C,D in the chart so that no letter appears twice in any new row, column, or diagonal.

Abstract Algebra and Puzzles [10/31/1998]

Do you have any ideas for projects that involve abstract algebra?

Adding and Multiplying to Get 7.11 [01/13/1999]

What four monetary values, when added or multiplied, equal $7.11? (Find A, B, C, D, such that A+B+C+D = A*B*C*D = 7.11.)

Ages 27-30 Puzzle [11/06/2001]

Alan was 27 years old the day before yesterday. Next year he will be 30. How is this possible?

Algebra Puzzle [04/09/2001]

If x+y+z = 1 and x^2 + y^2 + z^2 = 2 and x^3 + y^3 + z^3 = 3, what does x^4 + y^4 + z^4 equal?

Algorithm about Counting Letters and Words in Text [01/03/2007]

Comments and hints on where to find more information regarding an interesting algorithm related to texts in which by counting letters and words you always eventually wind up landing on the same word in the text. Martin Gardner has commented on the algorithm and used it in puzzles.

Algorithms for the Knapsack Problem [04/08/1998]

I need to find an algorithm to solve the Knapsack Problem, where a person needs to fit as many toys in his knapsack as possible.

Alphametric Problem [04/06/1999]

How do you solve a four-part alphametric where each letter represents a different digit?

Analyzing Prime Factors [06/14/2007]

Is there a number that has only three prime divisors (3, 5, and 7) and that has a total of 18 divisors?

Arithmetic Code [05/09/1999]

Break the code, given that each of the following is true in ordinary base ten arithmetic...

Arranging Numbers [06/26/2002]

Arrange the numbers 1-8 into the given configuration such that consecutive numbers are never adjacent.

Averaging 30 Miles Per Hour [03/17/1997]

If for half the distance of a trip you travel at 15 mph, what will your speed have to be for the rest of the trip for you to average 30 mph?

Bachet's Theorem [10/08/1998]

My maths teacher said that a mathematician discovered that you can make any number by adding together a combination of no more than 4 square numbers.

Bananas by Camelback [2/12/1995]

There is a camel that needs to travel 1000 miles to the nearest city. The camel has 3000 bananas but can only carry 1000 at a time...

Ben and Bill [12/21/1997]

Bill + Ben's age = 91. Bill is twice as old as Ben was when Bill was as old as Ben is now.

A Bidirectional Search [01/27/2003]

How can I get the numbers from 1-100 using only four 8's?

Birthday Calendar Puzzle [08/29/2001]

My question involves a game that I have played with my students for a long time, yet I am always unable to explain to them why the pattern works...

Birthday Probabilities [12/09/1997]

What's the minimum number of people you need in order for the probability that two of them were born on the same day of the week to be 50 percent?

Bouncing Cue Ball [10/29/1996]

A cue ball is launched at an angle of 45 degrees from the lower left corner of a pool table and ends up in the lower right corner. What rule will predict which corner the ball will hit? What patterns are involved?

Bowling Pin Problem [02/17/2006]

Ten bowling pins are set up in the usual way forming a triangle with the point facing the bowler. How can three pins be moved so that the point now faces away from the bowler?

Box A or Box B? [11/21/2000]

A professor shows you two boxes. Box A has $10K; Box B has $1M or $0. You can take Box B only or both boxes... Which do you choose?

Bulgarian Goats [05/17/2003]

How many goats are there in the herd? What are the sizes of the feeding groups once they have stabilised? Find at least two possible cyclic patterns of sizes.

Calendar Puzzle [09/09/2001]

Take a calendar with five weeks, choose only one day out of each week, and write the date of each day. Then add up all the dates. How can you know the total of all the dates without looking at the calendar?

Challenging Algebra Age Problem [02/29/2004]

A man has nine children whose ages are at an exact interval. The sum of the squares of the ages of each is the square of his own age. What is the age of each child and the man?

Challenging Puzzle Similar to the Four 4's [04/06/2006]

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

Change for a Dollar [03/07/1999]

How many different ways are there to make a change for a dollar?

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