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[math-fun] Decimal question
by David Wilson 30 Sep '06

30 Sep '06

Is there a finite decimal that uses the same digits as its reduced fractional representation?

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RE: [math-fun] A.I. question re ongoing chess championship match
by Daniel Asimov 29 Sep '06

29 Sep '06

Andy writes: << [I wrote]: > A.I. QUESTION: Is there a reasonable way, based on Kramnik's play > in previous games, to distinguish moves that he conceived solely on > his own from ones that are so brilliant that they were probably > computer-aided ? I don't think so. There could be evidence that Kramnik is *not* using computer assistance, if he's making fatal mistakes that a computer would not make. But if Kramnik were using a computer, and he knows there's a chance of getting caught, and he's smart, he won't just do whatever the computer recommends, because that might risk detection. Instead, he could just play his own game except in situations in which he felt that he had a close decision between two alternatives. In such a case, he could let the computer decide which play was better, and make that play. This would result in play indistinguishable to any human from Kramnik's play, that would nonetheless gain an advantage from use of the computer. >> Someone as smart as a top chessplayer would realize that even with illicit access to a good chess computer for use during a match, it can't be too obvious, and this alone may explain numerous suboptimal moves Kramnik has made. I don't want to believe he's cheating, but I must admit that even for fairly acute gastronomic distress or an overload of diuretics, 50 bathroom visits per game, if true, is suspicious. Even more so now that he's said he will not continue after today's game (which he lost, leaving the score at Kramnik 3, Topalov 2), because he has been barred from continued use of the players' private rest room, which is unmonitored. Seems to me Kramnik should be going out of his way to demonstrate that he is not using any kind of chess computer or access to one. --Dan

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RE: [math-fun] A.I. question re ongoing chess championship match
by Andy Latto 29 Sep '06

29 Sep '06

> -----Original Message----- > From: math-fun-bounces+andy.latto=pobox.com(a)mailman.xmission.com > [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On > Behalf Of Daniel Asimov > Sent: Thursday, September 28, 2006 10:23 PM > To: math-fun > Subject: [math-fun] A.I. question re ongoing chess championship match > > > Right now there is an ongoing chess championship, which (as followers > of the sport will know) has been accepted by both major camps > as the determiner of the next "world champion". > > (The match is to be 12 games, and is played in Elista, located close > to the middle of nowhere. After 4 games, Topalov has 1 point and > Kramnik has 3 points -- 2 wins & 2 draws.) > > Topalov has just signed a complaint saying that Kramnik is averaging > 50 bathroom breaks per game -- and the private, players' bathroom > is not monitored in any way. Topalov wants monitoring. (Cf. > < http://www.nytimes.com/2006/09/29/crosswords/chess/29chess.html >.) > Evidently Topalov is concerned Kramnik may be accessing a chess > computer for advice. > > A.I. QUESTION: Is there a reasonable way, based on Kramnik's play > in previous games, to distinguish moves that he conceived solely on > his own from ones that are so brilliant that they were probably > computer-aided ? I don't think so. There could be evidence that Kramnik is *not* using computer assistance, if he's making fatal mistakes that a computer would not make. But if Kramnik were using a computer, and he knows there's a chance of getting caught, and he's smart, he won't just do whatever the computer recommends, because that might risk detection. Instead, he could just play his own game except in situations in which he felt that he had a close decision between two alternatives. In such a case, he could let the computer decide which play was better, and make that play. This would result in play indistinguishable to any human from Kramnik's play, that would nonetheless gain an advantage from use of the computer. Andy Latto andy.latto(a)pobox.com

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[math-fun] Re: inconvenient directed graphs
by James Propp 29 Sep '06

29 Sep '06

I also posted my question to the domino forum (see jamespropp.org/about-domino) and got (1) a solution by Hugh Thomas and (2) a reference to the literature: H. Wielandt: Unzerlegbare, nicht negative Matrizen, (Engl.: Irreducible non-negative matrices) Math. Z. 52 (1950) 642-648. The optimal (or pessimal, depending on your point of view) matrices look like [0 + 0 0] [0 0 + 0] [0 0 0 +] [+ + 0 0] (as Dan Asimov observed). Thanks to all who responded (on- or off-line)! Jim Propp

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[math-fun] Tetrahedron puzzle
by Daniel Asimov 29 Sep '06

29 Sep '06

Let 0 < p < 2 and consider the graph of y = x^p in the xy-plane as well as the graph of z = y^q in the yz-plane for any q. In particular, consider the normal line to y = x^p at x=c, and the triangle T(p,c) that it makes with the nonnegative x- and y-axes. At the vertex (0,y0) of T(p,c) on the positive y axis, evaluate the function z = y^q and contruct a z-altitude A of height (y0)^q that's points parallel to the positive z-axis. Finally, let V(p,q,c) be the volume of the tetrahedron with T(p,c) as base and altitude A. Assume q = q(p) is chosen so that lim as c -> 0+ of V(p,q,c) exists and is positive. Call this limit L(p). PROBLEM: Find the maximum value of L(p) over all p in (0,2) to three decimal places. --Dan

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[math-fun] Q-factorials
by Schroeppel, Richard 29 Sep '06

29 Sep '06

I've been playing with Q-factorials: QF(0) == 1 QF(N+1) == QF(N) * (1 + Q + Q^2 + ... + Q^N). When Q=1, this gives QF(N) = N!. The degree of QF(N) is Triangle(N-1) = N(N-1)/2. Q-binomials seem to be polynomials: QB(A,B) = QF(A+B) / (QF(A) QF(B)). The degree of QB(A,B) is AB. The degree of QB(A,B,C) = QF(A+B+C) / (QF(A)QF(B)QF(C)) is AB+AC+BC. Etc. Perhaps we can use some of our standard bag of tricks for defining QF(N) for non-integer N. Put QF(Z) ~= Q ^ [ Z(Z-1)/2 ] for RealPart(Z) large, and divide by (Q^Z-1)/(Q-1) to go from Q(Z) to Q(Z-1). There's a set of functional equations similar to X! (X+1/2)! = (2X+1)! * stuff. Presumably there's also a reflection equation with QF(Z) QF(-Z) = more-stuff that looks like pi Z / sin(pi Z). Rich

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[math-fun] A.I. question re ongoing chess championship match
by Daniel Asimov 28 Sep '06

28 Sep '06

Right now there is an ongoing chess championship, which (as followers of the sport will know) has been accepted by both major camps as the determiner of the next "world champion". (The match is to be 12 games, and is played in Elista, located close to the middle of nowhere. After 4 games, Topalov has 1 point and Kramnik has 3 points -- 2 wins & 2 draws.) Topalov has just signed a complaint saying that Kramnik is averaging 50 bathroom breaks per game -- and the private, players' bathroom is not monitored in any way. Topalov wants monitoring. (Cf. < http://www.nytimes.com/2006/09/29/crosswords/chess/29chess.html >.) Evidently Topalov is concerned Kramnik may be accessing a chess computer for advice. A.I. QUESTION: Is there a reasonable way, based on Kramnik's play in previous games, to distinguish moves that he conceived solely on his own from ones that are so brilliant that they were probably computer-aided ? --Dan

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Re: [math-fun] Graph Question
by Michael Reid 28 Sep '06

28 Sep '06

> Is there a name for the operation which takes one graph to another, > such that the points of the new graph are the edges of the original > graph, and there is an edge between two such points if their > originating edges shared a point? I.e., each edge becomes a point, and > a point with valence n becomes C(n,2) edges. i believe the resulting graph is called the _line_graph_ of the original graph. i do not know if there's a specific term for the *operation* involved. mike

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[math-fun] Faro shuffling with jokers
by Marc LeBrun 28 Sep '06

28 Sep '06

It's fairly well-known that perfectly Faro shuffling a standard deck of playing cards 8 times will return them to their original order. For some reason I happened to compute Sloane A123320, which gives a table for different size decks cut at different places. I noticed that if you add two jokers to the deck and cut evenly the number is 52. If however you cut the regular deck evenly and place the jokers on top of the pile whose top card winds up on top it's still 8. On the other hand, if you do the same but put them under the pile whose bottom card winds up on the bottom it's 380. I can't think of any use for this factoid--other than maybe one magician hacking another (it'd require marks that can Faro shuffle!)--but perhaps it's still fun enough to share here...

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[math-fun] Graph Question
by franktaw＠netscape.net 28 Sep '06

28 Sep '06

Is there a name for the operation which takes one graph to another, such that the points of the new graph are the edges of the original graph, and there is an edge between two such points if their originating edges shared a point? I.e., each edge becomes a point, and a point with valence n becomes C(n,2) edges. Franklin T. Adams-Watters ________________________________________________________________________ Check Out the new free AIM(R) Mail -- 2 GB of storage and industry-leading spam and email virus protection.

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