Clik here to view.
Challenge #12 solution, part I
I’ll begin by providing an answer to the first of the three questions I posed in a previous post. To recap, the question in question went as follows: Given a positive integer n, in how many ways can n...
View ArticleClik here to view.
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...
View ArticleClik here to view.
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...
View ArticleClik here to view.
Math Teachers at Play #21
Math Teachers at Play #21 is up at Math Mama Writes…, and it includes this cute puzzle, which Sue apparently made up herself: The Numberland News runs personal ads. 21 was looking for a new friend and...
View ArticleClik here to view.
P vs NP: What’s the problem?
As promised (better late than never), I’m going to begin explaining the (in)famous P vs NP question (see the previous post for a bit more context). As a start, here’s a super-concise, 30,000-foot...
View ArticleClik here to view.
The chocolate bar game: losing positions in binary
Recall the chocolate bar game from my last post, whose winning and losing positions can be visualized like this: Here’s a list of some losing positions on or above the main diagonal (dark blue squares...
View ArticleClik here to view.
The chocolate bar game: losing positions characterized
The evident pattern from my last post is that losing positions in the chocolate bar game appear to be characterized by those where the binary expansion of is the same as the binary expansion of with...
View ArticleClik here to view.
The chocolate bar game: losing positions proved
In my last post I claimed that the losing positions for the chocolate bar game are precisely those of the form (or the reverse), that is, in binary, positions where one coordinate is the same as the...
View ArticleClik here to view.
Efficiency of repeated squaring
As you probably realized if you read both, my recent post without words connects directly to my previous post on exponentiation by repeated squaring Each section shows the sequence of operations used...
View ArticleClik here to view.
Efficiency of repeated squaring: proof
My last post proposed a claim: The binary algorithm is the most efficient way to build using only doubling and incrementing steps. That is, any other way to build by doubling and incrementing uses an...
View ArticleClik here to view.
Efficiency of repeated squaring: another proof
In my previous post I proved that the “binary algorithm” (corresponding to the binary expansion of a number ) is the most efficient way to build using only doubling and incrementing steps. Today I want...
View ArticleClik here to view.
The wizard’s rational puzzle (solutions, part 2)
At long last, here is the solution I had in mind for the Wizard’s rational puzzle. Recall that the goal is to figure out the numerator and denominator of a secret rational number, if all we are allowed...
View ArticleGoldilogs and the n bears
Once upon a time there was a girl named Goldilogs. As she was walking through the woods one day, she came upon a curious, long house. Walking all round it and seeing no one at home, she tried the door...
View ArticleClik here to view.
A new counting system
0 = t__ough 1 = t_rough 2 = th_ough 3 = through So, for example, trough through tough though though. English is so strange.
View ArticleClik here to view.
Clik here to view.
A few words about PWW #30
A few things about the images in my previous post that you may or may not have noticed: As several commenters figured out, the th diagram (starting with ) is showing every possible subset a set of...
View ArticleClik here to view.
Clik here to view.
Hypercube offsets
In my previous posts, each drawing consisted of two offset copies of the previous drawing. For example, here are the drawings for and : You can see how the drawing contains an exact copy of the...
View Article
