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# Category Archives: solutions

## The Steinhaus-Johnson-Trotter algorithm

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 algorithm, change, permutations, ringing, SJT, swap
9 Comments

## Fun with repunit divisors: more solutions

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 repunit

## Cassini’s identity

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

## Area paradox unmasked

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

## Triangular number equations via pictures: solutions

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 number, pictures, proof without words, triangular
10 Comments

## The broken weight problem: solutions and further exploration

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 balanced, broken, puzzle, ternary, weight
6 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 algorithm, binary, Calkin-Wilf, Euclidean, Haskell, hyperbinary, tree
6 Comments

## More hyperbinary fun

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 hyperbinary, induction
12 Comments

## Challenge #12 solution, part II

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 binary, hyperbinary
15 Comments

## Distributing cookies: solutions

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 binomial coefficients, combinatorial identities, combinatorics, cookies, counting, distribution
4 Comments