tag:blogger.com,1999:blog-11295132.post7849779898876842752..comments2007-10-26T07:41:34.247-07:00sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-11295132.post-38368261893079380312007-10-09T10:53:00.000-07:002007-10-09T10:53:00.000-07:00sigfpe, I was wondering if Djinn would find the natural isomorphisms without needing to be told that they were isomorphisms, but now I think that's unlikely. Asked to find maps between T^7 and T it would probably just find obvious canonical projections and embeddings. :-(<BR/><BR/>PS: your blog rejects the &lt;sup&gt; tag, which seems unnecessarily paranoid.Jeremy Hentynoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-84427790559439038612007-10-09T07:14:00.000-07:002007-10-09T07:14:00.000-07:00George,<BR/><BR/>Nicely done! I believe that is the shortest solution.<BR/><BR/>See also Cale's: http://cale.yi.org/autoshare/pennies.pngsigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-62125458207253414262007-10-09T07:10:00.000-07:002007-10-09T07:10:00.000-07:00** Spoiler Alert **<BR/><BR/>Here is a nice symmetrical solution in 18 moves. I'm pretty sure no shorter solution exists.<BR/><BR/>0 0 0 0 0 0 0 1 0<BR/>0 0 0 0 0 0 1 0 1<BR/>0 0 0 0 0 1 0 1 1<BR/>0 0 0 0 1 0 1 1 1<BR/>0 0 0 1 0 1 1 1 1<BR/>0 0 1 0 1 1 1 1 1<BR/>0 1 0 1 1 1 1 1 1<BR/>1 0 1 1 1 1 1 1 1<BR/>1 1 0 2 1 1 1 1 1<BR/>1 1 1 1 2 1 1 1 1<BR/>1 1 1 1 1 2 0 1 1<BR/>1 1 1 1 1 1 1 0 1<BR/>1 1 1 1 1 1 0 1 0<BR/>1 1 1 1 1 0 1 0 0<BR/>1 1 1 1 0 1 0 0 0<BR/>1 1 1 0 1 0 0 0 0<BR/>1 1 0 1 0 0 0 0 0<BR/>1 0 1 0 0 0 0 0 0<BR/>0 1 0 0 0 0 0 0 0George Bellhttp://www.geocities.com/gibell.geo/pegsolitaire/index.htmlnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-2361663478361773882007-10-08T21:40:00.000-07:002007-10-08T21:40:00.000-07:00jeremy,<BR/><BR/>Djinn will find functions with the right type, but not isomorphisms.sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-42293467563712452802007-10-08T21:39:00.000-07:002007-10-08T21:39:00.000-07:00George,<BR/><BR/>If you allow negative pennies then whether you realise it or not you're proving that two polynomials lie in a certain ideal of a polynomial ring! :-)sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-20850149115450635602007-10-08T21:19:00.000-07:002007-10-08T21:19:00.000-07:00Porges, there's a Haskell program called Djinn that creates Haskell expressions given only the type. See http://lambda-the-ultimate.org/node/1178 . Maybe it's powerful enough to discover these isomorphisms?Jeremy Hentynoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-43582536238382520882007-10-08T19:57:00.000-07:002007-10-08T19:57:00.000-07:00Interesting puzzle!!? I have an alternate proof thet doesn't rely on complex numbers, but it less slick.<BR/><BR/>Is 18 moves the shortest possible solution?<BR/><BR/>You can do it in only 4 moves if you allow "negative pennies".George Bellhttp://www.geocities.com/gibell.geo/pegsolitaire/index.htmlnoreply@blogger.comtag:blogger.com,1999:blog-11295132.post-78038161289431563682007-10-06T19:45:00.000-07:002007-10-06T19:45:00.000-07:00It would be very cool if someone came up with a program to automatically generate type isomorphisms such as these :)Porgeshttp://www.blogger.com/profile/02727258157936734796noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-64630908492139411882007-10-05T15:26:00.000-07:002007-10-05T15:26:00.000-07:00Brent,<BR/><BR/>Steal away!sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-69204213880068450492007-10-05T14:03:00.000-07:002007-10-05T14:03:00.000-07:00Hey, I'm planning to write a post on the "nuclear pennies" game on my math blog for high school students (mathlesstraveled.com). Mind if I steal your images? =)Brent Yorgeyhttp://www.mathlesstraveled.com/noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-32496017783537772012007-10-01T11:41:00.000-07:002007-10-01T11:41:00.000-07:00Another amazing paper along similar lines is Tom Leinster and Marcelo Fiore's <A HREF="http://arxiv.org/abs/math/0212377v1" REL="nofollow">Objects of Categories as Complex Numbers</A>, which shows how the "meaningless computation" of Blass' section 2 can be made rigorous and proves a general soundness theorem for such proofs.Mileshttp://www.blogger.com/profile/07136909835648629963noreply@blogger.com