Category Archives: sequences

Post without words #5, explained

If you stared for a while at the images in my previous post, you probably noticed some patterns, and maybe you even figured out some sort of rule or algorithm behind them. Commenter Yammatak expressed it as “You split it … Continue reading

Posted in pattern, pictures, posts without words, sequences, solutions | Tagged , , , , , | 3 Comments

Fibonacci multiples, solution 1

In a previous post, I challenged you to prove If evenly divides , then evenly divides , where denotes the th Fibonacci number (). Here’s one fairly elementary proof (though it certainly has a few twists!). Pick some arbitrary and … Continue reading

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences | Tagged , , , | 5 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 , , | 4 Comments


It is easy to generalize number bracelets to moduli other than 10—at each step, add the two previous numbers and take the remainder of the result when divided by m. Here are some pretty pictures I made of the resulting … Continue reading

Posted in arithmetic, fibonacci, iteration, pattern, pictures, sequences | 6 Comments

Number bracelets

Recently I’ve been volunteering with the middle school math club at Penn Alexander, a PreK-8 school in my neighborhood. Today we did (among other things) a fun activity I’d never seen before, called “number bracelets”. The students seemed to enjoy … Continue reading

Posted in arithmetic, iteration, pattern, sequences, teaching | Tagged , , , | 12 Comments

The hyperbinary sequence and the Calkin-Wilf tree

And now, the amazing conclusion to this series of posts on Neil Calkin and Herbert Wilf’s paper, Recounting the Rationals, and the answers to all the questions about the hyperbinary sequence. Hold on to your hats! The Calkin-Wilf Tree First, … Continue reading

Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions | Tagged , , , , , , | 6 Comments

Hyperbinary conjecture seeking proof for a good time, long walks on the beach

Here’s the latest progress on the hyperbinary sequence. We’re trying to figure out the inverse relation of the function : given a particular number , where does it occur in the hyperbinary sequence? That is, what are the values of … Continue reading

Posted in challenges, pattern, people, proof, sequences | Tagged , , , | 4 Comments