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[math-fun] CNN headline re unlikely gambling industry problem
by The Purple Streak, Hilarie Orman 10 Apr '03

10 Apr '03

cnn.com, travel section, today, Casinos hit by uncertainty Hilarie

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[math-fun] (HSM) Donald Coxeter
by Ed Pegg Jr 10 Apr '03

10 Apr '03

Sad news. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Coxeter.html Dear mathematical colleagues of Donald Coxeter, It is with great sadness that I have to tell you that on March 31, 2003 Donald passed away peacefully at is home in Toronto at the age of 96. His daughter Susan was at his side as she has been since the death of his beloved wife Rien. Respecting Donald's wishes, there will be no funeral or memorial service, but cremation with only son and daughter present. Family requests not to send any flowers but instead to reflect upon Donald's long and productive life. Rather than dwell in sadness, his death should be a catalyst for people to remember the wonderful things in life and enjoy the life the way he would have wanted it. Susan would like to share with you a recent photo of Donald with his youngest great-grandchild. The bumper sticker encourages us to celebrate Donald's life. I lost a mentor, we lost a colleague, the world lost a mathematician. Asia Ivic Weiss

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Re: [math-fun] tic tac toe variants
by Dan Hoey 08 Apr '03

08 Apr '03

Roland Silver <rollos(a)starband.net> writes: > ... I published a paper on the automorphisms the board (permutations > of the cells which preserve 4-in-a-row) some years ago (Amer. Math. > Monthly; Vol. 74, No. 3 (March 1967), pp. 247-254). Aside from the > obvious subgroup of 48 rotations and reflections, there are two > other independent automorphisms. One scrambles the board by > interchanging each of the 3 parallel pairs of inner planes, while > the other inverts the board by interchanging each of the 3 pairs > consisting of an inner plane and its parallel outer neighbor. > The whole group has 192 elements. I enjoyed that paper quite a bit. I especially appreciated the work you did to show that there were no _other_ symmetries. Later on, I was working with "Rubik's Revenge", the 4^3 Rubik's cube. I think it was David Singmaster who noticed that it had some extra symmetries, and I realized with amazement that they were the same as the extra symmetries of 4^3 tic-tac-toe. (These symmetries only work if you consider the puzzle to consist of an array of cubes all the way through, where the inner cubes have to be solved as well as the outer ones.) These symmetries--inverting some layers and permuting the noncentral layers--extend to larger sizes and dimensions of Rubik's cubes and tic-tac-toe boards. I suspect these are all there are, no matter how big you get, but I haven't a proof. I wonder if these groups have any other applications? Dan Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] transcendental puzzles
by Richard Schroeppel 07 Apr '03

07 Apr '03

A simple puzzle that leads to a transcendental equation: I'm going to cut a circular cookie into thirds (3 equal areas) by making two vertical cuts. Where should they be made? (I learned this from Gosper, but it must predate the invention of trigonometry. Probably just after cookies were invented.) A 3D version: The density of ice is .9, water is 1. A spherical iceberg of diameter 1 is floating in the ocean. How high is the tip of the iceberg over the surface of the water? Rich rcs(a)cs.arizona.edu

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[math-fun] "Masonry"(?) tilings
by R. William Gosper 07 Apr '03

07 Apr '03

Reply-to: rwg(a)osots.com Ed, I read a mistranslation of Nob Yoshigahara's forthcoming puzzle collection and wrote a Macsyma function to count rectangular 3x1 tilings that are not the abutment of two smaller such tilings. This must be trite stuff, but you might check out www.ippi.com/rwg/tatami.htm just to be sure. E.g., I was surprised that there's only one unsymmetrical 9x9. With both smaller and larger sizes, unsymmetrical solutions predominate. Also, the two completely symmetrical 11x15s might be manually discoverable. Copy whatever you find interesting-- that website is only for temporary stuff. --Bill Anagrams: ironstone-serotonin, precipitate-peripatetic, pictures-piecrust-cuprites-crepitus

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[math-fun] RE tic-tac-toe variants
by Guy Haworth 05 Apr '03

05 Apr '03

See www.icga.org ... Game-specific info ... Go-moku family re Go-Moku, Renju and Pente. Some variants of the 'Five in a Row' family have been 'solved' by computer. 4*4*4 ... Qubic ... is also a domain that evinced an interesting 'solution method' by computer. Not that the 'solved status of some games need subtract from the human enjoyment of them. guy

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Re: [math-fun] transcendental puzzles
by asimovd＠aol.com 04 Apr '03

04 Apr '03

Michael Kleber writes: << Mathematica vs Dan Asimov: [I wrote:] << PUZZLE: Given that x^2 + log_10 (2x) = 3, solve for x > 0 in closed form [most of Mathematica session omitted here] 5 Log[10] Out[2]= {{x -> Sqrt[---------]}} Log[100] Aah, much better. I have no idea why it fails to recognize Log[10]/Log[100]. >> OK, I'm impressed. It essentially solved it, though I agree it's mysterious why it screwed up with the last little simplification. (Perhaps a repeated Simplify[], or a FullSimplify[], would do the trick?) I wonder what kind of algorithm it used to solve such an offbeat equation. (The answer, as you've all realized by now, is x = sqrt(10) / 2.) --Dan

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Re: [math-fun] transcendental puzzles
by asimovd＠aol.com 04 Apr '03

04 Apr '03

As a teen, I made up the PUZZLE: Given that x^2 + log_10 (2x) = 3, solve for x > 0 in closed form (which I think I mentioned on math-fun a long time ago). Although it's not hard to solve for x explicitly, it seems to require human intuition rather than any algorithm. How would computer algebra systems handle this??? Since I don't have one handy, would someone out there be willing to use Macsyma, Maple, or Mathematica and ask it for the solution to this equation (without giving the software any prompts other than that x is real) ? I'm very curious to know what happens. --Dan

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Re: [math-fun] tic tac toe variants --- please
by asimovd＠aol.com 04 Apr '03

04 Apr '03

Thane asks for << a way to make tic-tac-toe, or something like it, into an interesting game Here are a number of tic-tac-toe-like games which are all more sophisticated and fun than your basic always-can-draw game. 1. Go-moku. This is played on a theoretically infinite square grid (a large piece of graph paper, or the beach, will do nicely). Each player marks any unoccupied square with their mark (X or O), marks are unchangeable, and the first player to get 5 in a row of their mark (vertically, diagonally, or horizontally) wins. (There are numerous variants. Cf. <http://project.blink-182.org/seb/spec.html> . However, be warned that most websites explaining the rules of the game seem to be written by sub-imbeciles.) 2. 3D tic-tac-toe. This is usually played on a 4x4x4 cube-shaped structure with four transparent 4x4 squares arranged one above the other with space to insert either of 2 kinds of markers. It's often available commercially, or you can build your own. First player to get 4 in a row right-left, front-back, up-down, in a planar diagonal, or in a main diagonal of the 4x4x4 cube wins. 3. 4x4 toral tic-tac-toe. Just like regular tic-tac-toe but played on a 4x4 board, and it takes 4 in a row in a straight line or diagonal to win. The main difference (besides 4 replacing 3) is that the winning line may wrap around, as though the game board has its opposite edges identified to make a torus. This means that instead of the expected 10 winning lines, there are now 6 more kinds of "diagonals" making 16 winning lines in all. (These 2 patterns, flipped and rotated, give all 6 new "diagonals".) ___________ | X | | X | | X | | X | ------------------- ___________ | X | | X | | X | |X | ------------------ 3.5 Of course 2. and 3. could be combined to get 3D toral tic-tac-toe -- but only for the advanced player! 4. Set. This commercial card game is somewhat like tic-tac-toe conceptually, but not mechanically. Each card has a graphic with 4 properties (C=color, N=number, S=shape, and L=line-style). The deck of 81 cards has one of every possible combination. The dealer shuffles the deck and not-too-quickly turns one card after another over and places it on the table (or floor). A "set" is any 3 cards having the following property: For each property P (= C, N, S, or L), either all 3 possibilities for P occur among the 3 cards, or else only 1 of the 3 possibilities occur. Any number 0 <= k <= 4 of the 4 properties can occur in all 3 possibilities, and then the remaining 4-k properties must each be constant among the 3 cards in the "set". (This may sound complicated at first, but it is easy to get used to on a gut level. (A set is really an affine line in the vector space (Z_3)^4, in case you really wanted to know -- and that's why it's like tic-tac-toe.) Anyhow, the first one to recognize a "set" among the face-up cards, yells "Set!" and collects the cards as a triple, over the course of the game stacking up the harvested "sets" so they can be counted when the deck has been all turned over (at which point there will be some unsettable collection of cards left). The one with the most "sets" wins. Set can be quite addictive. (An interesting problem discussed here some years ago and solved by Rich is this: What is the largest possible number of cards from the Set deck which has no "set" among them ?) --Dan

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[math-fun] April is National Math Awareness Month
by The Purple Streak, Hilarie Orman 03 Apr '03

03 Apr '03

Three days to the wind, and what have you done to celebrate? My contribution to Math Awareness Month was not to shout out the answer when a local public official fell apart in mid-sentence over 1 1/2 of 4. I suggest National Math Compassion Month, instead. Or have March be National Arithmetic Compassion Month and let April be National Math Fun Month. Hilarie

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