# The CTK Exchange

Epimenides (c. 7 century B.C.) once made a long pilgrimage to meet Buddha. When he finally met him, Epimenides said, "I have come to ask a question. What is the best question that can be asked and what is the best answer that can be given?"

Buddha replied, "The best question that can be asked is the question you have just asked, and the best answer that can be given is the answer I am giving."

Raymond Smullyan

*5000 B.C. and Other Philosophical Fantasies*

St. Martin's Press, NY, 1983

CTK Exchange is a page devoted to questions and answers concerning my pages and math in general. Several reasons led me to starting this page:

- From time to time visitors ask me questions I think might be of general interest.
- Sometimes I do not know answers and hope that somebody will come up with the right idea.
- Yet at other times an onlooker may have an idea that would enrich the question or the answer or both.

Thus every one can suggest and post either questions or answers on this page subject to one condition. I welcome your participation.

Please use the new form-based message board to post your questions and answers. All forums there are moderated. Even very urgent messages(**!**) will be answered in turn if at all.

Below is the archive of the pre-board exchange.

At the bottom of each question/answer page appear some or all of the following buttons:

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You are welcome to submit your remarks, observations or solutions to any problem. Difficult and unsolved problems are marked with Problems I believe to be on a research level are marked with |

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Exchange pages use JavaScript and JavaScript cookies. In case your browser is incapable of handling JavaScript just send me e-mail indicating your question and what it refers to. Thanks.

- An arithmetic puzzle (with digits 1-9)
- Re: An arithmetic puzzle (with digits 1-9) by Gary Darby

- Optimization in orthogonal matrices
- Multitasking mechanism
- Keplerian analysis of the volume of a wine barrel
- Re: Keplerian analysis of the volume of a wine barrel by Rod McFarland

- Partition of point sets
- Pythagorean Theorem in Literature
- Inside a spherical mirror
- Flipping pancakes and counting
- Re: Flipping pancakes and counting by David L. Arnett
- Re: Flipping pancakes and counting by Don Bowen
- Re: Flipping pancakes and counting by Jacob Mandelson

- A cryptarithm
- Re: A cryptarithm by David L. Arnett

- 5 circles covering
- Find the day
- Sequencing integers
- Probability in integer sequences
- Analog gadget name - trammel?
- Random chewing
- Connect 12 dots
- Re: Connect 12 dots by Tal Lachmann
- Re: Connect 12 dots by Thinh Nguyen
- Re: Connect 12 dots Jacob Mandelson

- Re: Connect 12 dots by Tal Lachmann
- A trigonometric inequality
- Base Calculator
- Nim Explained
- Three glasses of water
- Four 4s
- The Game of Nim
- Uniqueness of number 30
- What's a triangle?
- Hour Glass puzzle
- 4 Travellers problem
- Four Travellers problem
- Super numbers
- Arithmetic
- Cutting pipes
- Old ball puzzle
- Number Patterns
- Proof of Pythagorean Theorem #??????
- Four consecutive integers
- Factoring polynomials
- Biographies
- Integration in Spherical Coordinates of a cube
- Rounding numbers
- A simply looking problem
- Re: A simply looking problem
- Re: A simply looking problem by Christopher A. Pellegrino
- Re: A simply looking problem by Ted Carmichael

- Re: A simply looking problem by Christopher A. Pellegrino

- Re: A simply looking problem
- Probability Paradoxes
- Statistics
- Pythagorean Theorem
- The Central Limit Theorem
- Double negatives
- Re: Double negatives
- Re: Double negatives
- Re: Double negatives by Bonnie Warmund Teboul

- The terms of mathematics
- Pythagorean triples
- More on Pythagorean triples
- More on Pythagorean triples, II
- Cycloids
- Human Calculator
- Re: Human Calculator
- Re: Human Calculator by Steve Kaplan

- Re: Human Calculator
- Sliding puzzles
- Randomness
- Diagonal Method
- Trigonometric identity
- Negative powers
- Trigonometry in a triangle
- Distance to the horizon
- Social Choice
- Puzzles and Things
- Ellipse
- Question related with Hilbert Curve
- PMI
- Area of an infinite rectangle
- Incidence of birthdays
- Fibonacci sequence
- Using Pythagorean Theorem
- 40-60 steps
- Magic Square
- 3 problems of antiquity
- Mathematical induction
- Day of the week
- Linear Programming
- Checkmarks
- Magic Star
- Areas in a trapezoid
- Hanoi Tower
- Number series
- x
^{x}=10^{100}- Re: x
^{x}=10^{100}by David Radcliffe

- Re: x
- Geometry question
- Probabilities of decks of cards
- Bananas
- Counting on a grid
- Circle cutting
- Moebius strip dimensions
- Square
- gcd
- Kids' ages
- Pigeonhole principle
- Re: Pigeonhole principle
- Re: Pigeonhole principle by W.McWorter

- Hanoi Tower
- Dimensional Analysis
- Distance on the sphere
- Re: Distance on the sphere
- Re: Distance on the sphere by Ben Heath

- Guest book
- What the heck?
- Re: What the heck?
- Re: What the heck? by Richard Stockford
- Re: What the heck? by Ivan Sovic

- Re: What the heck? by Richard Stockford

- Re: What the heck?
- Moebius from ellipse
- 5x5 magic square
- Parabola and its focus
- Base converter page
- Diagonals
- Commas in numerals
- A sequence
- Googol
- What is Mathematics?
- Math teasers with 1001
- Double optimization
- Chinese remainder theorem
- A diophantine equation
- Intersections, intersections
- Synthetic division
- What is discrete mathematics?
- Random variable is defined...
- Approach to logic problems
- Number base equation
- How to win the lottery (sort of)
- Re: How to win the lottery (sort of)
- Re: How to win the lottery (sort of) by Christopher A. Pellegrino

- Re: How to win the lottery (sort of)
- Computing Fibonacci sequence
- Cauchy's limit
- Point inside a triangle
- terminology question
- Cauchy's inequality
- Advice on Nim
- Formula for the height
- Devil's games
- Names for the Powers of Ten
- Do parallel lines meet at infinity?
- Probabilities with two die
- Sphere and cone
- J.J.Sylvester's puzzle
- Multi-stage Monty Hall problem
- Asymptotes of a rational function
- Orthocentre
- On Bertrand's Paradox
- Coincidences in card dealing
- Minimum distance between objects in 2D
- Human Calculators, an Encore
- Binary floating point division
- Pay phone problem
- Probability terms
- Algebraic equations in quaternions
- Strange number sequence
- Cantor's triadic set
- Probability of losing keys
- Measurements in volume
- HOW TO CALCULATE AN ARCH
- How many tournaments?
- Re: How many tournaments?, by William A. McWorter Jr.

- Rock and Roll puzzle
- Sum of no numbers
- .9999... = 1?
- Base 36
- Doors and Graphs
- Why quadratic?
- Train Wheels Quandary
- Zero probability
- Getting wet
- Primes In Nature
- What's Next?
- Matter of Age
- Division by Zero
- English words without vowels
- Re: English words without vowels
- Re: English words without vowels by Skip Granberry
- Re: English words without vowels by Jonathan Matte
- Re: English words without vowels by Jacob
- Re: English words without vowels by Graham Cleverly

- Re: English words without vowels
- Are there math secrets?
- Dimensions of a line
- No nickel puzzle
- Is Zero a Whole Number?
- Unreal (complex) numbers
- Ancient divergent series
- Cantor's diagonal method
- Monty Hall dilemma
- i
^{i}and de Moivre's formula - Addition
- Falsity implies anything.
- Different answer
- Isoperimetric property of regular n-gons
- Integration Constant
- Angle Trisection
- Why does using reciprocals to divide fractions work?
- Crossing the river with dogs
- Re: Crossing the river with dogs by Patti Sprague

- Gender probabilities
- What is a number?
- Probability of birthday matches
- Is 1+1=2 an axiom?
- Hating mistakes and carrying bubbles
- Why there are 360 degrees in a circle?
- What do they mean?
- An infinite flippage of coin
- Properties of Pythagorean Triples
- Why squaring is so ubiquitous?
- Re: Why squaring is so ubiquitous?
- Re: Why squaring is so ubiquitous?, Graham Cleverly

- Re: Why squaring is so ubiquitous?
- Fibonacci Magic
- Fundamental Theorem of Calculus
- Division by Zero
- rotating a needle 360 degrees
- sums & products
- Function graphs in various number systems
- Moore-Penrose Inverse
- Math problems and puzzles
- The ceiling function
- Digits in Base 47
- Is 0 nothing?
- Re: Is 0 nothing?
- Re: Is 0 nothing? by Cale Gibbard
- Re: Is 0 nothing? by Sasha Goldshtein

- Re: Is 0 nothing? by Cale Gibbard

- Re: Is 0 nothing?

Copyright © 1996-2018 Alexander Bogomolny