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# Category Archives: challenges

## Making connections

Here’s something fun I was playing around with today. I generated 100 random points and connected some of them with lines. Can you figure out how I chose which lines to draw? How about these? (Same points, different lines.) Or … Continue reading

## Permuting permutations

As you probably know, there are ( factorial) different ways to put the numbers from through (or any set of distinct objects) in a list. For example, here are the different lists containing the numbers through : Each such list … Continue reading

## Blockly

It seems that Google is developing a graphical programming language called Blockly, inspired by Scratch but web-based, with the ability to compile down to JavaScript, Dart, or Python (or raw XML, so you can process it further). I can’t say … Continue reading

## Picture this

Picture this! is a very cool interactive thingy, made by Jason Davies, intended to get students (or anyone, really) thinking about some interesting math. Go play around with it and see if you can answer any of the listed questions … Continue reading

Posted in challenges, links, pattern, pictures
Tagged interactive, picture, rectangles, this
6 Comments

## Fibonacci multiples

I haven’t written anything here in a while, but hope to write more regularly now that the semester is over—I have a series on combinatorial proofs to finish up, some books to review, and a few other things planned. But … Continue reading

Posted in arithmetic, challenges, fibonacci, number theory, pattern
Tagged divisibility, fibonacci
12 Comments

## Fun with repunit divisors

In honor of today’s date (11/11/11), here’s a fun little problem (and some follow-up problems) I’ve seen posed in a few places (for example, here is a very similar problem). If I recall correctly, it was also a problem on … Continue reading

Posted in arithmetic, challenges, modular arithmetic, number theory, primes
Tagged divisors, primes, repunit
16 Comments

## A Fibonacci pattern

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences
Tagged fibonacci, number, pattern
4 Comments

## An area paradox

Here’s a fun paradox which has been around for quite a while and was apparently a favorite of Lewis Carroll. As you can verify for yourself, the two figures above are composed of two different rearrangements of the same four … Continue reading

## Triangular number equations via pictures

The other day I was fiddling around a bit with triangular numbers. By only drawing pictures I was able to come up with the following triangular number equations, where denotes the th triangular number (that is, the number of dots … Continue reading