Ask Dr. Math

High School Archive

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

Selected answers to common questions:
    Four-color map theorem.
    How many handshakes?
    Squares in a checkerboard.
    Tournament scheduling.
    Venn diagrams.

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Graph Theory [09/29/2001]

Why is a graph with five vertices, each having a degree of 3, impossible?

Graph Without Crossing Lines [7/19/1996]

There are three houses and three utilities: how do you connect each of the houses individually to the three utilities without crossing your lines?

Greatest Integer Functions [09/27/1998]

Can you help me solve for the graph of [y]=[x], where [] is the greatest integer function?

How Many are in the Group? [10/17/1996]

Everyone in the group had been to at least one of the parks...

How Many Balls Would Be Used? [5/8/1995]

Consider a knock-out tournament, say tennis or ping-pong, with n participants. The winner of any game goes on to the next round and the loser retires...How many balls have been used in the tournament?

How Many Distinct Patterns? [01/15/2001]

Given a large equilateral triangle divided into four smaller equilateral triangles, if two edges are painted white and the rest are painted black, how many distinct patterns are possible?

How Many Factors? [7/14/1996]

How do you find the number of factors for a number?

How Many Games in the Tournament? [01/15/2002]

There are eight teams in a single-elimination tournament. Each team gets to play until it loses. How many games will be played in the tournament?

How Many Threes? [06/12/1999]

If all the numbers from 1 to 333,333 are written out, how many times will the digit 3 be used?

Infinity Hotel Paradox [09/15/1999]

How can a hotel with an infinite number of rooms, all already occupied, accommodate the passengers of an infinite number of buses without doubling them up?

Integer Solutions of ax + by = c [04/03/2001]

Given the equation 5y - 3x = 1, how can I find solution points where x and y are both integers? Also, how can I show that there will always be integer points (x,y) in ax + by = c if a, b and c are all integers?

Inverse, Product of Permutations [04/27/2002]

I don't understand how to calculate the inverse or the product of permutations.

Josephus Problem [04/18/2003]

Every other person at a table is eliminated until there is only one person left. Who is the survivor?

Karnaugh Maps [05/07/2000]

What are Karnaugh maps? How are they used?

Knights of the Round Table [07/01/1998]

If x knights are sitting at a round table, and every other one is removed, who is the last one left sitting at the table?

The Königsberg Bridge [5/20/1996]

Do you have information on Konigsberg's bridge?

Lattice Points and Boundary Lattice Points [08/30/1998]

What is an interior lattice point and a boundary lattice point of a given shape (triangle, circle, rectangle, etc.)?

Line Drawn through Lines Puzzle [10/18/2001]

Given a box made up of 16 lines, with two rectangles above and three squares below, draw a line through each line without crossing any line twice.

Lines determined by 5 points [11/13/1994]

How many lines are determined by 5 points, no three of which are collinear?

Locker Problem [11/21/1997]

There are 1,000 lockers numbered from 1 through 1,000. The first student opens all the doors; the second student closes all the doors with even numbers...

Math Games Involving Forcing an Opponent into an Outcome [06/19/2004]

A very challenging math game provides the background for a discussion of how to find the winning strategy in 'reduced state' games, where players attempt to force a final outcome after a series of moves.

Math Logic - Determining Truth [04/13/1999]

A number divisible by 2 is divisible by 4. Find a hypothesis, a conclusion, and a converse statement, and determine whether the converse statement is true.

Matrix Multiplication [12/18/1998]

Why does matrix multiplication work? Why are the rows multiplied and added with the columns?

Meaning of '-ominoe' [11/07/2001]

We are drawing pictures of dominoes, triominoes, tetrominoes, and pentominoes. What is the meaning of the root "ominoe"?

Minimum Number of Cuts to Slay the Dragon? [01/14/2007]

A magic dragon has 3 heads and 3 tails. A knight with a magic sword can make four types of cuts--one head, two heads, one tail, or two tails. For each type of cut, the dragon regrows one head, nothing, two tails, or one head respectively. The knight must remove all heads and tails to slay the dragon. What's the fewest number of cuts he can make?

Moving Knights on a Chessboard [01/27/1999]

Given 4 knights at the 4 corners of a 3-by-3 chessboard, can the knights exchange places if they can move only in the following way?

Multi-Dimensional Four-color Theorem [08/08/1997]

Has any work been done on theorems like the four-color theorem for different dimensions?

The N-Color Theorem? [07/27/2002]

What happens if we try to generalize the Four Color Theorem to other numbers of dimensions?

New School Lockers [01/28/2001]

Which locker was touched the most?

Nim [09/26/2000]

What is the principle of Nim and what is its application?

No Three Red Beads Together [09/16/2001]

Given 10 beads on a necklace, 6 white and 4 red, how many ways can the beads be arranged so that no three red beads are together?

Number/Color Cube [09/13/2001]

You want to make a number cube by putting the numbers 1,2,3,4,5,6 on the face. 1/5, 3/6, and 2/4 must be on opposite faces. Each face is a different color. How many ways can you make the cube?

Number of Unordered Partitions [08/18/1999]

Is there a formula for the number of unordered partitions of a positive integer p(n)?

Number of Ways to Move [1/30/1996]

I have a group of squares which together form a larger square. In how many ways can you travel from the upper left corner of the large square to the lower right corner by only going down or to the right?

Number Systems: Two Points of View [06/30/1998]

What are the number systems?

Number Theory Proofs [06/24/1999]

How can I prove that the equations (x,y) = g and xy = b can be solved simultaneously if and only if g^2|b for integers g, b?

Objects in a Pyramid [7/8/1996]

Objects are stacked in a triangular pyramid... how many objects are in the nth layer from the top?

Occupancy Problem [08/06/2001]

Given n bins and m (indistinguishable) balls, how many arrangements are possible such that no bin has greater than r balls?

Odd and Even Vertices [1/30/1996]

We are trying to trace networks without crossing a line or picking up our pencils, but how can we know if a vertex is odd or even?

Odd Number of Hands, Even Number of People [08/31/2001]

Every person on earth has shaken a certain number of hands. Prove that the number of persons who have shaken an odd number of hands is even.

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