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Selected answers to common questions:
Four-color map theorem.
How many handshakes?
Squares in a checkerboard.
- Rat Population [04/27/1999]
Estimate the number of offsping produced from a pair of rats in one
- Recursive and Explicit Formulas [01/19/1999]
Is there an easy way to convert recursive formulas into explicit ones and
- Resources for NIM [07/03/1997]
Where can I find information about the game of NIM?
- Reversed Digits Theorem [06/24/1999]
For a positive integer abc..., if (abc...)^n = xyz... and if
(a+b+c+...)^n = x+y+z+..., how can I prove that (...cba)^n = ...zyx?
- Rock, Paper, Scissors [03/29/2001]
If three people are playing Rock-Paper-Scissors, how many different
combinations can be made, assuming order doesn't matter?
- Rubik's Cube Combinations [04/11/2001]
I read that a rubics cube has 4 quintillion different possible
combinations. Is this number correct? How can I calculate this value on
- Segmenting Paths [08/20/1998]
A path between opposite vertices of the square is made up of hundreds of
horizontal and vertical segments. What is the best approximation to the
length of the path - 24, 34, 44, or more than 44?
- Set and Element Relations [03/23/2003]
On a set of n elements, how many relations are there that are
reflexive and antisymmetric? irreflexive and symmetric?
- Set Equality [10/12/1998]
Can you help me show that (A-B)-C = (A-C)-(B-C), where A, B, and C are
- Sets and Integer Pairs [6/10/1996]
A) Prove that the sum of a specified element of one set is greater than
or equal to a specific number (n + 1)/2; B) Find all the ordered pairs of
integers (m, n) that satisfy the equation (n^3 + 1) / (mn - 1).
- The Seven Bridges [8/28/1996]
What is the problem from the 1700s about a town with seven bridges, where
you want to cross each bridge exactly once?
- Showing Divisibility [07/12/1998]
How do you show that 5^(2n) + 3(2^(2n+1)) is divisible by 7?
- Simple Proof by Induction [08/27/1999]
How can I show by mathematical induction that the proposition "if n >= 1
then 3n >= 1 + 2n" is true?
- Solving a System of Modular Equations with Multiple Variables [12/15/2004]
Thoughts on solving a system of modular equations such as:
(1919ab) mod 5107 = 1
1919(a+1)(b-1) mod 5108 = 5047
1919(a+2)(b-2) mod 5109 = 1148
1919(a+3)(b-3) mod 5110 = 3631
1919(a+4)(b-4) mod 5111 = 2280
- Subsets of Real Numbers and Infinity [08/22/2001]
Am I correct in saying that both the whole number set and the integer set
have an infinite number of numbers within them, and therefore are of the
- Summation by Parts [01/07/2004]
Using 'E' to represent sigma, is there an approximate solution to
E(Ai*Bi) = ? where i = 0,1,...,n if Ai is known explicitly and E(Bi)
- Sum of Reciprocals [08/05/2008]
Find seven unique positive integers such that the sum of their
reciprocals is 1.
- Sum of Squares of Two Odd Integers [10/26/1999]
How can I prove that the sum of the squares of two odd integers cannot be
a perfect square?
- Symmetric, Transitive, and Reflexive Relations [11/10/1998]
Suppose R is a symmetric and transitive relation on A, and for each a in
A there is b in A such that (a,b) and is in R. Show that R is an
- System-Level Programming and Base 2 [05/03/2001]
In computer programming, I have a result that contains several values,
always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304,
how can I tell if 2^3 exists in 304?
- System of Equations and Gauss-Jordan [11/29/1998]
Solve using the Gauss-Jordan method: a 5-percent solution of a drug is
mixed with 15- and 10-percent solutions...
- Three Houses, Three Utilities [07/15/1999]
Can you solve it using 2 dimensions? How?
- Three Number Theory Questions [10/25/1999]
Find the sum of the digits in 4444^4444; find how many times the digit 1
occurs from 1 up to 10,000,000,000; find 3 integers greater than 5^100
that are factors of (5^1985)-1.
- Tiling with Dominoes [08/06/2001]
A 6-square by 6-square board is tiled completely with 18 2x1 dominoes.
Prove that at least one horizontal or vertical line can be drawn along
the edges of the dominoes that divides the board into 2 regions, without
cutting any dominoes in half.
- Total Membership [08/20/1999]
At a country club 35 people play golf, 28 swim, and 24 play tennis. Of
these, 6 play golf and tennis only, 9 play golf and swim only, and 7 play
tennis and swim only. 8 people do all three. How many members are there
- 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 [05/24/2001]
Is there an easy solution to the "Traveling Salesman Problem"?
- Traveling Through a Square [11/25/2001]
How do I get from the bottom left-hand corner of a 64-block square to the
top right-hand corner, only going through each square once?
- Triangle in Randomly Colored Plane [10/28/2002]
Prove: Assume that all points in the real plane are colored white or
black at random. No matter how the plane is colored (even all white or
all black) there is always at least one triangle whose vertices and
center of gravity (all 4 points) are of the SAME color.
- 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
- Types of Variables [01/14/1999]
Can you explain the different types of variables, such as free variables
and bound variables?
- Unions and Intersections: Proving Sets [10/17/1999]
How can I verify a proof of the statement A - (B union C) = (A - B)
intersect (A - C)?
- Unknown Numbers and a Venn Diagram [11/26/2001]
The GCF of two numbers is 20 and the LCM is 840. One of the numbers is
120. Explain how to find the other number and use the Venn diagram method
- Using Graph Theory to Count Routes [07/15/1998]
How do you use a diagram to count the number of different direct routes
that connect five cities?
- Using Trees [10/18/1996]
What are trees used for? What are some examples?
- Venn Diagram - Choose One of Three Options [01/24/1999]
Members of a computer class choose at least one of three options. How
many are taking just one? ... Use a Venn diagram.
- Venn Diagram: Goops, Gorps, Gorgs [09/19/2002]
Every Goop is a Gorp. Half of all Gorgs are Gorps. Half of all Gorps
are Goops. There are 40 Gorgs and 30 Goops. No Gorg is a Goop. How
many Gorps are neither Goops nor Gorgs?
- Venn Diagram of Natural Numbers [09/22/1999]
How can I construct a Venn diagram comparing the numbers 1 through 100 in
these 4 areas: odd, even, composite and prime?
- Venn Diagram of Our Number System [12/13/2002]
I don't know how to include complex numbers that consist of a real
part and an imaginary part. Can you please diagram this for me?
- Venn Diagram to Classify Quadrilaterals [01/02/2003]
I am looking for a Venn diagram that will accurately display the
relation among trapezoids, parallelograms, kites, rhombi, rectangles,