http://bit-player.org/2006/first-bites An amateur's outlook on computation and mathematics. Wed, 23 Jul 2008 19:13:13 +0000 http://wordpress.org/?v=1.5.2 http://bit-player.org/2006/first-bites#comment-922 Tue, 05 Sep 2006 18:31:37 +0000 http://bit-player.org/2006/first-bites#comment-922 [...] 2D scaling in certain games Recently Friedman and Landsberg’s experimental results on certain scaling properties in the game chomp has been featured in Science News. (See another take on their work at bit-player.) It has been noted that the classic combinatorial game nim seems to have a “rougher”, easier to analyze structure than the “smooth” structure of more complicated games like chomp. In a similar vein, an interesting tidbit is that the limit exists for all integer choices of , , and if is the function giving the value of two-roweosd chomp positions; whereas if is the value of two nim heaps, the only values of which work for all , are the powers of two. So, I thought it would be interesting to graph the difference between and for these different functions. There are at least two different ways of taking the difference between two nim-values: simply interpret them as integers and subtract one from the other, or use what is here the more natural operation of nim addition, conveniently provided directly by most CPU’s and implemented in C-style languages as the operator “^” - that’s right: binary XOR. Of course, what would be really interesting is whether this kind of scaling behavior extends to more than two dimensions, but I figured that as long as I am getting pretty pictures for 2D scaling, I have a duty to share them. First up is a look at two heap nim, compared using XOR. Click on the image to download a MPEG movie showing the box with x and y up to 256, for . Here black is zero difference, and white is maximum difference in that frame. Notice the many scaling properties associated with the powers of two in both space and time. I believe I have seen this basic idea used in screensavers before, which is appropriate since it is extremely fast. [...] […] 2D scaling in certain games Recently Friedman and Landsberg’s experimental results on certain scaling properties in the game chomp has been featured in Science News. (See another take on their work at bit-player.) It has been noted that the classic combinatorial game nim seems to have a “rougher”, easier to analyze structure than the “smooth” structure of more complicated games like chomp. In a similar vein, an interesting tidbit is that the limit exists for all integer choices of , , and if is the function giving the value of two-roweosd chomp positions; whereas if is the value of two nim heaps, the only values of which work for all , are the powers of two. So, I thought it would be interesting to graph the difference between and for these different functions. There are at least two different ways of taking the difference between two nim-values: simply interpret them as integers and subtract one from the other, or use what is here the more natural operation of nim addition, conveniently provided directly by most CPU’s and implemented in C-style languages as the operator “^” - that’s right: binary XOR. Of course, what would be really interesting is whether this kind of scaling behavior extends to more than two dimensions, but I figured that as long as I am getting pretty pictures for 2D scaling, I have a duty to share them. First up is a look at two heap nim, compared using XOR. Click on the image to download a MPEG movie showing the box with x and y up to 256, for . Here black is zero difference, and white is maximum difference in that frame. Notice the many scaling properties associated with the powers of two in both space and time. I believe I have seen this basic idea used in screensavers before, which is appropriate since it is extremely fast. […]

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