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Category Archives: recursion
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 applicable whenever we have some recursively … Continue reading
Posted in computation, games, recursion
Tagged coins, dynamic, game, optimal, play, programming, proof, recurrence, strategy, tree, twoplayer, zerosum
Comments Off on Computing optimal play for the greedy coins game, part 4
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 … Continue reading
Posted in computation, games, recursion
Tagged coins, game, optimal, play, proof, recurrence, strategy, tree, twoplayer, zerosum
2 Comments
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 quickly in … Continue reading
Posted in computation, games, recursion
Tagged coins, game, optimal, play, proof, recurrence, strategy, tree, twoplayer, zerosum
1 Comment
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 play … Continue reading
Posted in computation, games, recursion
Tagged coins, dynamic programming, game, optimal, play, proof, recurrence, strategy, twoplayer, zerosum
2 Comments
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 the resulting complexity shows how powerful selfreference is (for … Continue reading
Posted in arithmetic, recursion, sequences
Tagged difference, integers, Recamán, repeat, sequence
5 Comments
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 like the curvature , except that … Continue reading
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 three (forming what we called a “kissing set”). This … Continue reading
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 constructing them. Mostly I will state … Continue reading
Posted in geometry, pattern, pictures, recursion
Tagged Apollonian, gasket, recursion, tree
4 Comments
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 have created … Continue reading
Posted in arithmetic, links, pictures, primes, programming, recursion
Tagged diagrams, factorization, Haskell
15 Comments
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 how I did … Continue reading
Posted in arithmetic, pictures, primes, programming, recursion
Tagged diagrams, factorization, Haskell
72 Comments