tag:blogger.com,1999:blog-8398797088391606752.post8914760131413557079..comments2021-06-17T02:17:44.683-07:00Comments on PLEKTIX: Andi's Factor GameBen Allenhttp://www.blogger.com/profile/15594823641514744644noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-8398797088391606752.post-12061831607742457412019-07-02T08:41:45.573-07:002019-07-02T08:41:45.573-07:00I'm certain that the strategy can be concisely...I'm certain that the strategy <i>can</i> be concisely described. It's clear from the (very interesting) structure that this is one of the class of games that are isomorphic to Nim. The work, then, is to describe that isomorphism: for a given magic number M, what Nim-game N is it equivalent to?<br /><br />As a first step along those lines, it seems to be a multi-dimensional version of the chocolate-bar puzzle I think I first saw in an Ian Stewart book.<br /><br />First, write out the factorial decomposition: 12 = 2^2*3. The factors themselves don't really matter; just the powers.<br /><br />Next, imagine an n-dimensional block assembled from a bunch of n-cubes, where n is the number of prime factors. The size in each dimension is the power of the prime factor in the decomposition. One corner cube is marked.<br /><br />A move consists of "slicing" the block, and discarding the side not containing the marked cube. The player forced to take the marked block loses.<br /><br />The isomorphism should be pretty clear. This reduction should lay bare the real structure of each game; M=12 plays the same as M=20 plays the same as M=18, and so on. As such, it should be easier to describe the isomorphism from the block game to Nim.John Armstronghttps://www.blogger.com/profile/15177732626660057584noreply@blogger.com