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[math-fun] Tightly coloring K_n?
by Michael Kleber 25 Sep '16

25 Sep '16

Suppose you color the edges of K_n with C colors. Say the "spread" of a color = how many of the n vertices are an endpoint of an edge of that color. How small can you make the spread of the most-spread color? The pigeon-hole principle says you can't get below n/sqrt(k), since you'll have at least n-choose-2 / k edges of some color, and even if you pack them all into a complete subgraph, that's how big it would have to be. But surely you can't force *all* the colors into a really tight complete-graph-like arrangement, so seems like you should be able to improve on that substantially. Is there terminology for this question? My naive Google searchings are clobbered by the usual kind of edge coloring, where edges of the same color aren't allowed to meet at a vertex, so maybe I'm just looking for the right names to find prior results. --Michael -- Forewarned is worth an octopus in the bush.

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[math-fun] Question about base 3/2
by James Propp 25 Sep '16

25 Sep '16

Is it possible to write 1/3 as an infinite sum of the form a_1 (2/3)^1 + a_2 (2/3)^2 + ... where the sequence of nonnegative integer coefficients a_1, a_2, ... is eventually periodic? Jim Propp

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Re: [math-fun] Bar bet
by Dan Asimov 25 Sep '16

25 Sep '16

----- On Sun, Sep 25, 2016 at 12:01 PM, Hans Havermann wrote: >> "The volume of a regular tetrahedron (triangular pyramid with unit edges) >> is exactly half the volume of a square pyramid with unit edges." > > This is (I think) equivalent to stating that the volume of an octahedron is four times the volume of a tetrahedron (unit edges assumed). . . . At some point after the above time Andy Latto wrote: The geometrical "proof" of this is that you can assemble 4 unit tetrahedrons and a unit octahedron to produce a tetrahedron with edge length two, and therefore area equal to 8 tetrahedrons. ----- Lovely proof, Andy. Much better than doing the calculation (though it was fun doing it in my head). —Dan

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Re: [math-fun] Bar bet
by Bill Gosper 24 Sep '16

24 Sep '16

> From: Don Reble <djr(a)nk.ca> > To: math-fun(a)mailman.xmission.com > Sent: Friday, September 23, 2016 1:52 PM > Subject: Re: [math-fun] Bar bet > >> "The volume of a regular tetrahedron (triangular pyramid with unit edges) >> is exactly half the volume of a square pyramid with unit edges." >> True or false? --rwg > > Trick puzzle: prove that the volume of _every_ such pyramid > (unit edges, regular polygon base) is a quadratic surd. On 2016-09-23 13:59, Eugene Salamin via math-fun wrote: > The only remaining nontrivial case is the pentagonal base, and it's > clear that all the needed heights are no more involved than stuff > containing sqrt(5). In[445]:= PolyhedronData[{"Pyramid", 5}, "Volume"] Out[445]= 1/24 (5 + Sqrt[5]) > For the hexagonal base, the pyramid is flat, zero > volume, and for higher order polygons, the lateral edges cannot reach > the vertex while remaining of unit length. > > -- Gene In[587]:= PolyhedronData[{"Pyramid", 6}, "Volume"] During evaluation of In[587]:= PolyhedronData::notdef: PolyhedronData has no value associated with the specified argument(s). >> Why, do you suppose?-) --rwg

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[math-fun] Tenure-Track CS Position
by Michael Collins 24 Sep '16

24 Sep '16

My department is again advertising a full-time tenure-track position in computer science, in any area of specialization. Christopher Newport's CS department is an excellent place to work and I am happy to chat with anyone who might have questions. Mike Collins <http://cnu.edu/titleix-eo/pdf/8530%20assistant%20professor% 20of%20computer%20science.pdf>

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Re: [math-fun] Bar bet
by Don Reble 23 Sep '16

23 Sep '16

> "The volume of a regular tetrahedron (triangular pyramid with unit edges) > is exactly half the volume of a square pyramid with unit edges." > True or false? --rwg Trick puzzle: prove that the volume of _every_ such pyramid (unit edges, regular polygon base) is a quadratic surd. -- Don Reble djr(a)nk.ca

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[math-fun] Dario Alpern's QUAD
by Zak Seidov 22 Sep '16

22 Sep '16

Dear SF & MF gurus, plz HELP! Something is wrong with Dario Alpern's QUAD: https://www.alpertron.com.ar/QUAD.HTM They say that Java is obsolete(?) and I should do something that I don't understand:( I use Win XP and Win10 if it matters. Thanks, Zak Seidov

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Re: [math-fun] Dario Alpern's QUAD
by Axel Vogt 21 Sep '16

21 Sep '16

> From: Zak Seidov <zakseidov(a)mail.ru> > Subject: [math-fun] Dario Alpern's QUAD ... > https://www.alpertron.com.ar/QUAD.HTM > > They say that Java is obsolete(?) and I should do something > that I don't understand:( > I use Win XP and Win10 if it matters. The site says "This page uses JavaScript, ..." and after allowing that it works for me (Firefox 45), no need for Java.

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Re: [math-fun] Dario Alpern's QUAD
by Dave Dyer 21 Sep '16

21 Sep '16

Oracle has been making it harder and harder to deploy java applets, requiring real (not self-signed) certificates and changes to the tags associated with applets. Without recent maintenance, there's no chance applets will work. Also finding a browser which still supports java applets is getting harder - chrome and edge are completely out.

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Re: [math-fun] Dario Alpern's QUAD
by Dave Dyer 21 Sep '16

21 Sep '16

Oracle has been making it harder and harder to deploy java applets, requiring real (not self-signed) certificates and changes to the tags associated with applets. Without recent maintenance, there's no chance applets will work. Also finding a browser which still supports java applets is getting harder - chrome and edge are completely out.

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