tag:blogger.com,1999:blog-264226589944705290.post8808460967016671074..comments2023-11-05T03:45:25.001-08:00Comments on God Plays Dice: A puzzle about splitting up numbers into groupsMichael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-264226589944705290.post-49196372331332028032011-08-13T08:35:53.280-07:002011-08-13T08:35:53.280-07:00If {a_n} is any sequence of complex numbers such t...If {a_n} is any sequence of complex numbers such that sum |a_n| < infinity, then<br /><br />sum (-1)^{w(n)} a_n = <br />sum (-1)^{w(n)} (a_{2n} - a_{2n+1})<br /><br />where w(n) is the sum of the binary digits of n.<br /><br />It follows (by induction on k) that if p is a polynomial function of degree less than k, then<br /><br />sum [n=0 to 2^k - 1] (-1)^{w(n)} p(n) = 0.<br /><br />This solves the puzzle, but I wonder if there are other nice applications of this identity.Davidhttps://www.blogger.com/profile/09232747857608296294noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-25460774479309855842011-08-13T05:49:26.708-07:002011-08-13T05:49:26.708-07:00For a card trick application (with statistical ove...For a card trick application (with statistical overtones!) see:<br /><br />http://www.maa.org/columns/colm/cardcolm200712.html<br /><br />Also see A135419 at http://oeis.org/<br /><br />Array read by rows, showing the ways of splitting the numbers from 1 to 16 into two groups so that the numbers in each group have the same sum (68) and the same sum of squares (748)<br /><br />http://cardcolm.blogspot.com"Card Colm" Mulcahyhttps://www.blogger.com/profile/00320489161498127953noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-49414845863444906342011-08-12T23:25:21.651-07:002011-08-12T23:25:21.651-07:00This is the "Tarry-Escott" problem, also...This is the "Tarry-Escott" problem, also known as a "multigrade". <br /><br />Prouhet proved that your solution, based on Thue-Morse, works in general, back in 1851.Jeffrey Shallithttps://www.blogger.com/profile/12763971505497961430noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-3770669672466623532011-08-12T20:28:04.952-07:002011-08-12T20:28:04.952-07:00In fact, Rex, I didn't know that. Thanks!In fact, Rex, I didn't know that. Thanks!Michael Lugohttps://www.blogger.com/profile/01950197848369071260noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-66181938593467611772011-08-12T20:08:31.704-07:002011-08-12T20:08:31.704-07:00In case you weren't aware, the pattern of the ...In case you weren't aware, the pattern of the two sets follows the Thue-Morse sequence. See pages 4--6 of the following PDF: http://www.math.uga.edu/olympiad/10/teacher-team10.pdfAnonymousnoreply@blogger.com