Category Archives: solutions

The Steinhaus-Johnson-Trotter algorithm

Posted on January 3, 2013 by

In a previous post I posed the question: is there a way to list the permutations of in such a way that any two adjacent permutations are related by just a single swap of adjacent numbers? (Just for fun, let’s … Continue reading →

Posted in combinatorics, pattern, solutions | Tagged , , , , , | 9 Comments

Fun with repunit divisors: more solutions

Posted on November 17, 2011 by

In Fun with repunit divisors I posed the following challenge: Prove that every prime other than 2 or 5 is a divisor of some repunit. In other words, if you make a list of the prime factorizations of repunits, every … Continue reading →

Posted in arithmetic, iteration, modular arithmetic, number theory, primes, programming, proof, solutions | Tagged

Cassini’s identity

Posted on May 15, 2011 by

My previous post asked you to take any Fibonacci number, square it, and also multiply the two adjacent Fibonacci numbers, and see if a pattern emerged. Here’s a table I made for the first 6 Fibonacci numbers: (Hmm, the numbers … Continue reading →

Posted in algebra, fibonacci, induction, pattern, proof, solutions | Tagged , , | 8 Comments

Area paradox unmasked

Posted on May 8, 2011 by

In my last post I presented a paradox, where a set of four pieces forming an 8×8 square could apparently be rearranged to form a 5×13 rectangle, summoning an extra unit of area out of thin air. Quite a few … Continue reading →

Posted in geometry, pictures, puzzles, solutions | Tagged , , , | 1 Comment

Triangular number equations via pictures: solutions

Posted on April 14, 2011 by

Here are some solutions to my previous post. However, they are almost certainly not the only solutions! If you have other cool ways to visualize any of these (or any other triangular number equations) feel free to post in the … Continue reading →

Posted in arithmetic, pattern, pictures, proof, solutions | Tagged , , , | 10 Comments

The broken weight problem: solutions and further exploration

Posted on May 11, 2010 by

First of all, let me say to all my readers how fantastic it felt to post a puzzle, after not posting anything for two months, and get eighteen thoughtful, insightful comments in just three days; it’s every blogger’s dream. You … Continue reading →

Posted in arithmetic, challenges, number theory, solutions | Tagged , , , , | 6 Comments

The hyperbinary sequence and the Calkin-Wilf tree

Posted on October 18, 2009 by

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

More hyperbinary fun

Posted on September 28, 2009 by

When I originally posed Challenge #12, a certain Dave posted a series of comments with some explorations and partial solutions to part II (the hyperbinary sequence). Although I gave the “solution” in my last post, no solution to any problem … Continue reading →

Posted in challenges, induction, pattern, proof, recursion, sequences, solutions | Tagged , | 12 Comments

Challenge #12 solution, part II

Posted on September 19, 2009 by

Yes, that’s right, that Challenge #12, posted one year, five months, and a day ago. You see, I have this nasty habit of starting things and not finishing them… well, better late than never! Question two of the aforementioned challenge … Continue reading →

Posted in challenges, counting, induction, pattern, sequences, solutions | Tagged , | 15 Comments

Distributing cookies: solutions

Posted on April 14, 2009 by

And now for some solutions to the cookie distribution problem. I’m actually going to describe four different methods of solution, and thereby (re)discover some nice combinatorial identities along the way. This is what I love about combinatorics—you discover all this … Continue reading →

Posted in counting, proof, solutions | Tagged , , , , , | 4 Comments