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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Spiral Problem [02/03/2003]

Given an x,y plane, with the number zero at the center, so it has coordinates (0,0), if given the coordinates, is there a formula to find the number that will be at those coordinates?

Squares in Rectangle Formula [06/30/2003]

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

Squares of Positive Integers [8/13/1996]

The positive integers a and b: the numbers 15a+16b and 16a-15b are both squares of positive integers...

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?

Stack of Oranges [1/9/1995]

A stack of oranges is compactly arranged so the bottom layer consists of oranes in an equilateral triangle with n oranges on a side. The layer next to the bottom consists of n-1 oranges on a side. This pattern continues upward with one orange on the top. How many oranges are there?

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?

Stairs on an Escalator [05/21/2001]

A woman walks 10 steps down on a downward-moving escalator to reach the bottom. As she reaches the bottom, she runs back up the same escalator at a speed 5 times that which she walked down, covering 25 steps in reaching the top. How many steps are visible on the escalator when it is still?

Strategy for Playing Mastermind and Similar Games [03/26/2007]

I'm trying to guess a 4-digit number with unique digits and after each guess I'm told how many digits are correct and in the right place and how many are correct but in the wrong place. I use that feedback to make better and better guesses until I find the correct number. Is there a general strategy to minimize the guesses needed?

Subtraction Puzzle [08/18/2002]

For numbers A, B, C, and D, subtract A from B, (or vice-versa; you must be left with a whole number, not a negative one). Repeat with B and C, C and D, and D and A. After about 6 steps, you will always end up with 0000. The puzzle is to get as many steps as possible.

Sugar Cubes and Coffee Cups [09/26/2002]

Given 20 sugar cubes and 3 cups of coffee, how many cubes must be put into each cup of coffee if there have to be a total of an odd number of cubes in each cup?

Sum of First Four Digits Equals Units Digit [10/20/2001]

How many even five-digit numbers have the property that the sum of the first four digits is the units digit?

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 [02/04/2001]

What numbers can be expressed as the sum of a string of consecutive positive integers?

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?

Surface Area Of Three Cubes Glued Together [10/22/2002]

You have 3 cubes, one with an edge measuring 2, one 6, and one 8. If you glue them together so you have the smallest possible surface area, what will that surface area be?

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?

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 Fractions [09/06/2001]

Three fractions together add up to one. Place the nine digits 1-9 in the fractions to make the equation a/bc + d/ef + g/hi = 1 correct.

Three Hands on a Clock [07/03/2001]

At what time after 12:00, to the fraction of a second, are the three hands on a clock on top of each other?

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 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?

Tic-Tac-Toe on a Torus [03/29/2001]

Can you make a tic-tac-toe game that won't end in a tie?

Tiling a Mutilated Chessboard With Dominoes [08/29/2003]

Suppose we take an ordinary chess board and randomly remove a black square and a white square. Is it always possible to cover what remains with 2x1 dominoes? If yes, how? If no, why not?

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 [10/08/2000]

Can you prove the formula 2^n - 1 for the least number of moves it takes to move all n discs to another peg in Towers of Hanoi?

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?

Traveling Salesman Problem [12/3/1995]

Find a procedure that describes how to find the minimum distance between any 15 randomly placed dots.

Triangles within a Triangle [11/10/1996]

If multiple small equilateral triangles are drawn within a larger one, what is the relation between the number of small triangles lying on the base of the big triangle and the total number contained within the big triangle?

The Truel [10/13/1998]

A truel is a duel with three participants, rather than two. Whom should Mr. Black shoot first to survive?

Tumbling Dice by Robert Abbott [03/17/2002]

The center square is both the start and the goal: to solve the maze, you must move the die off the center square, then find a way to move it back onto that square...

TURKEYS puzzle [11/28/2001]

Count the number of ways you can trace the word TURKEYS in the triangular array. You may only move to one of the two letters directly below the letter you are on.

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?

Twenty Quadrilaterals from Nine Dots [04/04/1999]

How can you get 20 quadrilaterals from 9 dots?

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.

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