User urban - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:36:00Z http://mathoverflow.net/feeds/user/15231 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89053/does-a-generalized-queen-split-the-upper-p-positions-of-wythoff-nim-into-two-new Does a generalized Queen split the upper P-positions of Wythoff Nim into two new beams of P-positions? Urban 2012-02-20T22:06:10Z 2012-04-24T11:03:08Z <p>Wythoff Nim is an impartial game where 2 players take turns in reducing the heights of two finite heaps of tokens. Two types of moves are allowed </p> <p>(I) Remove any number of tokens from precisely one of the heaps, at least one and at most a whole heap. (These are the allowed moves in 2 heap Nim.)</p> <p>(II) Remove the same number of tokens from both heaps, at least one from each heap and at most the number of tokens in the smallest heap.</p> <p>The positions of an impartial game are partitioned into P and N. The second player to move wins if and only if the position is in P. The unique teminal position for both Nim and Wythoff Nim is empty heaps. </p> <p>The P-positions of 2 heap Nim are all configurations with the same number of tokens in both heaps. The P-positions of Wythoff Nim are of the forms $(\lfloor \phi n\rfloor, \lfloor\phi^2 n\rfloor),(\lfloor\phi^2 n\rfloor,\lfloor\phi n\rfloor), n\in {\bf Z}_{\ge 0}$, where $\phi = \frac{1 + \sqrt{5}}{2}$. </p> <p>It is clear that Wythoff Nim is an extension of Nim. By adjoining the type (II) moves to the game of Nim the unique "accumulation point" of P-positions of Nim has \emph{split} into two new accumulation points of P-positions of Wythoff Nim.</p> <p>We wonder whether this splitting of P-positions continues if we adjoin the following new type of (symmetric) moves to the game of Wythoff Nim. </p> <p>(III) Remove $t>0$ tokens from one of the heaps and $2t$ tokens from the other, provided the remaining heap sizes are non-negative.</p> <p>The initial (upper) P-positions of this new game (called (1,2)-GDWN) are $(0,0), (1,3), (2,6), (4,5),\ldots$. </p> <p>Experimental results gives four distinct "accumulation points" for the P-positions of this game (with upper convergents of ratios of heap sizes $1.478\ldots$ and $2.247\ldots$). It is known that the upper P-positions of the new game do not converge.</p> <p>It is not hard to prove that the non-terminal P-positions partition the positive integers. (Use that the type (I) and (II) moves are a subset of all moves.)</p> http://mathoverflow.net/questions/88773/branches-of-the-fibonacci-word-tree/88819#88819 Answer by Urban for Branches of the Fibonacci Word Tree Urban 2012-02-18T09:49:55Z 2012-02-18T09:49:55Z <p>This is interesting. Do you have a reference to "thin ∞-ominos"?</p> http://mathoverflow.net/questions/60358/structure-of-nonaveraging-sets-of-integers/60381#60381 Comment by Urban Urban 2012-08-22T15:26:08Z 2012-08-22T15:26:08Z L. Moser's construction from 1953 is perfectly constructive though. His sequence begins 100000, 1000100100, 1000400200, 1000900300, 1001600400, 1002500500, \ldots and becomes nearly as dense as Behrends: for sufficiently large $n$ it contains at least $n^{1-c/\sqrt{\log n}}$ numbers in an interval of length $n$, for some constant $c$. http://mathoverflow.net/questions/89053/does-a-generalized-queen-split-the-upper-p-positions-of-wythoff-nim-into-two-new Comment by Urban Urban 2012-07-10T16:45:24Z 2012-07-10T16:45:24Z I have recently proved a split. It is available at the arXiv. However the convergence properties of the splitted sequences remains a mystery. Included is an interesting Lemma bounding lower asymptotic density of P-positions lower heap sizes for any extension of Wythoff Nim to the inverse of the Golden section. http://mathoverflow.net/questions/88773/branches-of-the-fibonacci-word-tree/88819#88819 Comment by Urban Urban 2012-03-02T17:21:57Z 2012-03-02T17:21:57Z The flower petals within the sunflower's cluster are in a spiral pattern similar to this one. http://mathoverflow.net/questions/88773/branches-of-the-fibonacci-word-tree/88819#88819 Comment by Urban Urban 2012-03-02T17:11:20Z 2012-03-02T17:11:20Z <a href="http://en.wikipedia.org/wiki/File:SunflowerModel.svg" rel="nofollow">en.wikipedia.org/wiki/File:SunflowerModel.svg</a> http://mathoverflow.net/questions/88773/branches-of-the-fibonacci-word-tree/88819#88819 Comment by Urban Urban 2012-03-02T17:09:26Z 2012-03-02T17:09:26Z <a href="http://en.wikipedia.org/wiki/File:Helianthus_whorl.jpg" rel="nofollow">en.wikipedia.org/wiki/File:Helianthus_whorl.jpg</a> http://mathoverflow.net/questions/88773/branches-of-the-fibonacci-word-tree/88819#88819 Comment by Urban Urban 2012-02-19T16:05:13Z 2012-02-19T16:05:13Z Hi Johan, I hope I will earn overflow votes on my own merits soon :)