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September 21

Puzzle

I heard this a little while ago, and I don't think I've asked it here yet. I don't know the answer, and so far as I know neither does the guy who came up with it. Given a simple graph, choose a well-ordering of the points. Following that ordering, label each point with an ordinal number, subject to the following restriction: The label must be as small as possible, without being equal to the label of any adjacent point that's earlier in the ordering. The Grundy number of the graph is the supremum, in the ordinals, of the labels of all the graph's points, over all possible well-orderings of the graph. So for instance, the graph on the points A, B, C, and D with the lines AB, BC, and CD would have Grundy number 2, since that's the highest number you get in the graph from any ordering of its vertices. The possible labelings are 0101, 0120, 1010, and 0210. The question is, what's the Grundy number of the plane, where two points are adjacent if they're unit distance apart? I can get it up to the first uncountable ordinal, using a bit of recursion and some perturbation arguments, because I can produce any countable ordinal as a label of the graph, but I don't know if it's possible to specify an ordering that would produce an uncountable ordinal, or where you would go from there. Any ideas? Black Carrot (talk) 22:57, 21 September 2008 (UTC)[reply]

A related question, that may be easier: Will it be a limit ordinal? Black Carrot (talk) 22:58, 21 September 2008 (UTC)[reply]

Elliptic curves with isomorphic group structure that are not isomorphic

This is a continuation of a question I asked a few days ago. If two elliptic curves are isomorphic, it means more than just that their group structures are isomorphic. So, are there easy examples of two elliptic curves over the rationals that are not isomorphic but their group structures are known and are isomorphic? StatisticsMan (talk) 23:53, 21 September 2008 (UTC)[reply]

Consider

y^{2}=x^{3}+x\,;\,j=1728\,,

y^{2}=x^{3}-1\,;\,j=0\,.

Curves are not isomorphic as curves, because they have different values of j. But for both curves, rank is 0 and group of rational points is isomorphic to C₂; the only rational points on these curves are (0,0) for the first curve and (1,0) for the second (each curve group also includes a point at infinity, which is its identity element). Gandalf61 (talk) 10:52, 22 September 2008 (UTC)[reply]

Thank you once again. That is a very easy example. 129.186.52.59 (talk) 15:41, 22 September 2008 (UTC)[reply]