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[math-fun] Pandiagonal Sudoku?
by Christian Boyer 01 Apr '06

01 Apr '06

Do you know if a pandiagonal Sudoku has already been published? I have a 25x25 example, the smallest possible size for this problem, but perhaps the same work has already been done. "Pandiagonal Sudoku" means that all the diagonals and broken diagonals of the square should also contains all the numbers, as it is for rows, columns, and sub-squares. It is proved that a pandiagonal Latin square cannot exist for 2k and 3k orders, meaning that a pandiagonal Sudoku of standard size (9x9) is impossible. I know published 25x25 pandiagonal Latin squares, but they are not Sudokus, because they are not organized in 5x5 sub-squares. Christian.

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[math-fun] Sum of reciprocals of 8877690 integers with distinct digits
by Robert Baillie 30 Mar '06

30 Mar '06

Problem E2533 in the American Mathematical Monthly (1975 or 1976) says there are 8877690 integers that have distinct digits. The problem is to estimate the sum of their reciprocals. I thought I read someplace that someone had computed the exact sum. Does anyone know how big the numerator and denominator are? And, by the way, what is the approximate sum? Bob Baillie

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[math-fun] Road Sign Math
by Henry Baker 29 Mar '06

29 Mar '06

http://www.roadsignmath.com/

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[math-fun] Mathematics as Hypertext ... and a Logic Assistant
by Guy Haworth 29 Mar '06

29 Mar '06

I'm contemplating two projects for final-year CS students to do: 1) Thinking of mathematics as (partly comprising) a set of: Theorems with Preconditions and Proof(s), Generalisation theorem-theorem relationships, Theorem-(used in/by)-Proof relationships Counterexamples ... I see this information forming a database with an application 'projecting' this information to a web-interface which allows one to navigate around 'theorem space'. This seems well-suited to the web, and the opportunities of multimedia (graphics, animation, video, sound) is just a bonus - perhaps helped also by the XML-style language MathML. I'm thinking of asking a CS student to create such a system. Question is - is there anything out there of this kind already? 2) The intended system is a 'Logic Assistant'. A user-student would be trying to prove something using the formal means of a logic with already-known results and rules of inference. The system would either know the proof (as provided data) or be trying to work it out too [and I think there are systems for this 'out there' but I don't know about them]. The student would start with say 100 points and lose a few points for every 'atom' of hint provided by the system. The system would be able to manage hints, compare what the student proposed with the right answer, and general be an assistant/mentor/tutor in the learning process. Same question as for '1' really - is there anything out there of this kind already? Thanks in advance for any inputs - Guy Haworth, Univ of Reading, UK

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[math-fun] Mathematica Guidebooks ?
by Robert Baillie 27 Mar '06

27 Mar '06

Springer publishes a 4-volume set of Mathematica reference books by Michael Trott, at $80 per volume. They are described at http://mathematicaguidebooks.org Has anyone seen hard-copies of these? What do you think? Are they useful? Are they a waste of money? Bob Baillie

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[math-fun] F-Secure vs mfarch?
by R. William Gosper 27 Mar '06

27 Mar '06

I read mail over a link to unix, and have *never* had a virus. I de- installed Norton for its uncontrollable popups, surfing naked for a few days due to 15hr dialup download for F-Secure. Finally got it, thanks to Ed Hudson. First scan reported three unspecified viruses, then died in an infinite loop while reading mfarch/math-fun-archive-950622-960702 ! Really died. Not even killing the window killed the looping process. So I rebooted and just scanned mfarch. Two viruses reported, then the same death. I buzzed through the whole file with emacs. Nothing amiss. So I punted F-Secure and downloaded Panda, which ran ten times faster, deleted a bunch of cookies, and also excised the raw text of some classic viri from some ancient inboxes that I knew were "infected" and only open with emacs. (Norton, Trend, and F-Secure missed these, perhaps deliberately.) Panda did not attack the virus definitions presumably in the F-Secure download (which survived the deinstall) nor anything in mfarch. Does anyone else have F-Secure and mfarch? Do I have an incredibly subtle virus whose purpose is to embarrass F-Secure? --rwg DELUSIONAL OULED-NAIL'S ANDOUILLES

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[math-fun] Mandelbrot Set: the song
by James Propp 26 Mar '06

26 Mar '06

Check out the song "Mandelbrot Set" by Jonathan Coulton: http://www.jonathancoulton.com/lyrics/mandelbrot-set Sample of the lyrics: "Mandelbrot's in Heaven, at least he will be when he's dead / Right now he's still alive and teaching math at Yale..." (Would that make New Haven the opposite of Heaven?) ;-) Jim

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Re: [math-fun] Terminology
by dasimov＠earthlink.net 19 Mar '06

19 Mar '06

Franklin writes: << Ah, but it isn't that simple. Consider f, not as a function from X to Y, but just as a function. In set-theoretic terms, as a set of ordered pairs, such that nothing appears as the first element of any two distinct pairs. Now the domain is the set of elements appearing as the first element of any pair, and this is exactly the domain in any other sense. But the only reasonable definition for the range is the set of all elements appearing in the second element of any ordered pair; and this is the image in the other sense. Another way to look at it: if Z is the image of f: X -> Y, then f can be regarded as a function X -> Z (or X -> W for any W containing Y as a subset), and this is really the same function. So the image (often called the range) is a property of the function, while the "codomain" is a property of the function as viewed in a particular context, namely as a function X -> Y. >> I don't know about this. (For me, "just a function" always includes a domain & range, in the traditional terminology -- but I know what you're saying.) If every function had its codomain by definition shrunken down to its image, then every function would be onto, and questions about whether certain functions are, or could be, onto would become meaningless. But above all: In order to define a specific function, one needs to begin with a set to which the second element of its ordered pairs belong; it is not until after the function is defined that one can even ask what its image is. At that point, you have the option of defining a new function whose ordered pairs have their second elements in the image of the first function. But you also have the option of retaining the original such set (codomain) if there is some reason to do so. E.g., this makes it much easier to ask questions like this: Given a polynomial function of degree d, in n variables, with integer coefficients P: Z^n -> Z, characterize the set I(n,d) of all possible images. (I don't know whether this problem has been solved or not. Anyone know if it has?) --Dan

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Re: [math-fun] Terminology
by dasimov＠earthlink.net 19 Mar '06

19 Mar '06

Franklin writes: << Regarding defining a specific function, I am thinking in terms of ZF set theory. There, a function is just a set of ordered pairs (of sets) with the requisite uniqueness property. Such a function does not bring with it any concept of its codomain, just the image - which is normally called the range. The only general range would be the set of all sets - which does not exist. >> I would be interested to learn on what basis you state that ZF has a definition of function that agrees with what you write above. To the best of my knowledge the only kind of object in ZF is a set; the word "function" does not play a role. I am also not clear about what question your last sentence about "general range" is attempting to answer. --Dan

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Re: [math-fun] Home on the Range
by Henry Baker 19 Mar '06

19 Mar '06

Dan, I too noticed this back in the 1960's, when I compared my calculus (engineering-oriented) book with my algebra book. Unfortunately, the calculus textbook people haven't been particularly careful about this particular issue -- it's only a matter of time before some major standardized test gets this particular thing wrong & creates more embarrassment for the mathematics community. Before condemning the calculus people too severely, it would be good to check historical usage -- I'm not sure that the algebra people would win the historical usage argument (even among mathematicians). After all, "range" for a gun or an airplane is the actual achievable distance, not the entire shooting range, or the list of all airports in the world. At 08:17 PM 3/18/2006, dasimov(a)earthlink.net wrote: >When I first learned the rigorous definition of a function f: X -> Y, >the terminology was unequivocal: X is the domain of f, and Y is the >range of f. The latter is unconditionally the case, irrespective of whether >f is onto. > >Nowadays, so many high school course and ill-considered calculus texts have >redefined "range" to mean f(X) -- what I was taught is called the image of f -- >that in teaching I've leaned away from using the word range to mean either thing. > >Now I call the classical range by the term "codomain" (and I still call image "image"). > >It's a shame when the cognoscenti feel obliged to follow the ignorami. > >--Dan

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