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Computing optimal play for the greedy coins game, part 4
Last time I explained a method for computing best play for instances of the greedy coins game, which is feasible even for large games. This general approach is known as dynamic programming and is...
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Computing optimal play for the greedy coins game, part 3
In a previous post we saw how we can organize play sequences in the greedy coins game into a tree. Then in the last post, we saw how to work our way from the bottom of the tree upward and compute the...
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Computing optimal play for the greedy coins game, part 2
I want to explain in more detail how we can think about computing the best possible score for Alice in the greedy coins game, assuming best play on the part of both players. I glossed over this too...
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Computing optimal play for the greedy coins game
Recall the greedy coins game, in which two players alternate removing one of the coins from either end of a row, and the player with the highest total at the end is the winner. What if we wanted to...
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The Recamán sequence
I recently learned about a really interesting sequence of integers, called the Recamán sequence (it’s sequence A005132 in the Online Encyclopedia of Integer Sequences). It is very simple to define, but...
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Apollonian gaskets and Descartes’ Theorem II
In a few previous posts I wrote about “kissing sets” of four mutually tangent circles, and the fact that their signed bends satisfy Descartes’ Theorem, (Remember that the signed bend of a circle is...
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Apollonian gaskets and Descartes’ Theorem
In my previous post, I explained a recursive procedure for drawing Apollonian gaskets. Given any three mutually tangent circles, there are exactly two other circles which are mutually tangent to all...
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Apollonian gaskets
In my last post I showed off this tantalizing picture: This pattern of infinitely nested circles is called an Apollonian gasket. Over the next post or two I’ll explain some cool math behind actually...
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More factorization diagrams
My post on factorization diagrams from a month ago turned out to be (unexpectedly) quite popular! I got ten times as many hits as usual the day it was published, and since then quite a few other people...
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Factorization diagrams
In an idle moment a while ago I wrote a program to generate "factorization diagrams". Here’s 700: It’s easy to see (I hope), just by looking at the arrangement of dots, that there are in total. Here’s...
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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...
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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...
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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...
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Recounting the Rationals, part III
First, a quick recap: continuing an exposition of the paper Recounting the Rationals, we’re investigating the tree of fractions shown below (known as the Calkin-Wilf tree), which is constructed by...
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Recounting the Rationals, part II (fractions grow on trees!)
Today I’d like to continue my exposition of the paper “Recounting the Rationals”, which I introduced in a previous post. Recall that our goal is to come up with a “nice” list of the positive rational...
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