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Category Archives: solutions
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 CalkinWilf 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 CalkinWilf Tree First, … Continue reading
Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions
Tagged algorithm, binary, CalkinWilf, 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
Chessboard counting: solutions and further challenges
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 chess, chess board, counting, rectangles, squares
2 Comments
Predicting Pi: solution
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 approximants, approximation, floor, pi, sequence
5 Comments
Challenge #12 solution, part III
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