Category Archives: solutions

Area paradox unmasked

Posted on May 8, 2011 by Brent

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 Brent

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 Brent

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 Brent

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 Brent

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 Brent

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 Brent

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

Chessboard counting: solutions and further challenges

Posted on March 30, 2009 by Brent

And now for some solutions to the chessboard counting challenges. The first challenge was to count the number of squares of any size on an 8×8 chessboard. The key here (as with many counting problems) is to break the problem … Continue reading →

Posted in challenges, counting, solutions | Tagged , , , , | 2 Comments

Predicting Pi: solution

Posted on October 23, 2008 by Brent

Now for the solution to the question in my previous post, which asked what you can learn about , given the sequence of integers . Nick Johnson commented: Well, the obvious thing one can learn given just |(10^n)r| is the … Continue reading →

Posted in convergence, pattern, sequences, solutions | Tagged , , , , | 5 Comments

Challenge #12 solution, part III

Posted on April 24, 2008 by Brent

And now for the solution to problem #3 from Challenge #12, which asked: how many ways are there to write a positive integer n as a sum of powers of two, with no restrictions on how many powers of two … Continue reading →

Posted in counting, pattern, recursion, sequences, solutions | 1 Comment