CTK Insights

For a prime , two integers are both divisible by with the probability , because this only happens when the two integers have the residue 0 (one out of available residues) modulo . Two integers are mutually prime if they have no common nontrivial factors, prime facors in particular. Assuming divisibility by one prime is [...]

In their textbook Discrete Algorithmic Mathematics authors Stephen Maurer and Anthony Ralston offer a light-hearted example [pp. 217-221]: When checking into a hotel nowadays, sometimes a guest receives a four-digit "combination", not a key. On the door of each room is a keypad. Any time those four digits are entered in the proper sequence (regardless [...]

Every book carries an ISBN or EAN-13 or both numbers. ISBN or EAN-13 are two ways to encode information that would allow to identify a book on the market. ISBN contains 10 digits EAN-13 thirteen. All but the last digit encode country, manufacturer (publisher), and product (book) information. The last digit serves as a safeguard [...]

What is the setup? You'll need a set of cards with numbers 1, ..., N written one per card; N not too small and not too big. Say, for N = 8: How to start? Shuffle the cards somehow and place them in a row: What is the activity? Look at the number on the [...]

Today's activity is based on a problem by Vyacheslav Proizvolov offered at the 1985 All-Union Soviet Math Olympiad. What do you need? Strictly speaking, that activity needs nothing beyond a piece of paper and a pencil. However, it may be convenient to have numbers, say, 1 through 20, written on small paper pieces: What is [...]

The setup Draw 6 dots more or less evenly distributed over a circle: The activity Join the dots by lines of two colors, say, red and blue. Join all pairs of dots. See how many lines of the two colors emanate from each dot. What's to observe At every dot there are at least three [...]

The setup A student is presented with a small pile of objects. The task To split a pile into two; count the number of objects in each, and compute the product of the two numbers. This step is repeated with any of the present piles until the only remaining piles each contain a single object. [...]

Counting a group of objects can be done in many different ways. The most fundamental idea is that counting is at all possible in the sense that, regardless of the manner in which it is performed, the result is always the same. For example, place random numbers in a rectangular array and then compute separately [...]

Proofs Without Words is a great educational device that helps students understand and teachers convey mathematical facts. Professor Roger Nelson of Lewis & Clark College has a special knack for the PWW; a rare issue of Mathematics Magazine comes out without one if his creations. The latest (June 2011) is no exception. What do you [...]

I have recently come across an article by Atara Shriki of the Technion - Israeli Institute of Technology - where she extended an engaging property of the graph of y = x³ introduced by R. Honsberger. At an arbitrary point P on the graph of y = x³ draw the tangent line and mark its [...]