July 2016 - mathfun@mailman.xmission.com

[math-fun] A problem about finite sets
by Victor Miller 10 Sep '16

10 Sep '16

Say that a finite set of integers, A, has *property P* is there are two non-empty disjoint subsets S and T of A, such that sum(S) = sum(T) (where sum(X) is the sum of the elements of X). 1) Show that every 10 element subset of {1,...,100} has property P (this is fairly easy). 2) Is it true that every 9 element subset of {1,...,100} has property P? Victor

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[math-fun] Reentrant polygons with area zero
by James Propp 16 Aug '16

16 Aug '16

For which n>2 does there exist a reentrant n-gon whose vertices are evenly spaced on a circle and whose signed area (aka algebraic area) is zero? When n is even, there is an easy solution: draw a circuit from 1 to 2 to 3 to ... to n/2-1 to n/2 to n to n-1 to n-2 to ... to n/2+1 to 1 (here I'm numbering the n vertices cyclically). For n=3, it's trivially impossible to find such an n-gon, and I've checked by brute force (using Mathematica) that there's no such n-gon when n=5 or n=7. For n=9, there is such an n-gon (which I found through brute-force search): http://mathenchant.org/nonagon.pdf Mathematica assures me that 1/2 (-1/2 Sqrt[3] sin(\[Pi]/18)+1/2 sin(\[Pi]/9)-sin((2 \[Pi])/9)-2 sin(\[Pi]/18) sin((2 \[Pi])/9)-1/2 cos(\[Pi]/18)-1/2 Sqrt[3] cos(\[Pi]/9)+2 cos(\[Pi]/18) cos((2 \[Pi])/9)+2 sin(\[Pi]/9) cos(\[Pi]/9)) simplifies to 0, which implies the claim. Is there a nice (preferably trig-free) way to prove it? That is: is there a nice way to show that the total area of the three scalene black triangles equals the total area of the four nonscalene black triangles, maybe using dissection and area-preserving linear transformations, or maybe using algebraic tricks involving roots of unity? I suspect that when n is prime, there's no such polygon. Can any of you find a proof? Also, can any of you construct such a polygon with n=15? It'd be nice if the answer to my original question were "Precisely when n is composite". Jim Propp

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[math-fun] Brains patterns for solving math problems
by Hilarie Orman 14 Aug '16

14 Aug '16

I hope this line of research leads to improvements in education: http://www.nytimes.com/2016/07/29/science/brain-scans-math.html Looks like "encoding" really heats up the grey matter. Hilarie

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[math-fun] Fw: Re: Reentrant polygons with area zero
by Adam P. Goucher 31 Jul '16

31 Jul '16

(Oops, this was meant to go to the mailing list!) > Sent: Sunday, July 31, 2016 at 4:40 PM > From: apgoucher(a)gmx.com > To: "James Propp" <jamespropp(a)gmail.com> > Subject: Re: [math-fun] Reentrant polygons with area zero > > Consider the space V of rational linear combinations of pth roots of unity. This space has dimension p - 1, since the degree of the minimal poly of omega is p - 1. > > It's also the direct sum of the spaces Re(V) and Im(V). Each of these has dimension at most (p - 1)/2, and equality must hold since dim(V) = dim(Re(V)) + dim(Im(V)). > > The result follows from dim(Im(V)) = (p - 1)/2. > > Best wishes, > > > Adam P. Goucher > > -----Original message----- > Sent: Sunday, 31 July 2016 at 13:30:31 > From: "James Propp" <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Reentrant polygons with area zero > It's a nice approach, but I'm missing a step near the end. Why is it true > that > > When n is prime, the *only* integer linear relationships between > > the imaginary parts of roots of unity are generated by relationships > > of the form: > > > > Im[w^k] + Im[w^-k] = 0 > > > ? > > Note that Adam is talking about the *nontrivial* roots of unity. After > all, the relation Im[w^0] = 0 has an odd number of terms. But this doesn't > mar Adam's argument because w^(a_0 - a_1), ... are all unequal to 1 (which > is I suppose what Adam was getting at by saying "Note that none of these > terms are zero"). > > Jim Propp > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] Reentrant polygons with area zero
by Bill Gosper 31 Jul '16

31 Jul '16

(Alternate Subject: Twisted STOP signs--permutations of the vertices of a regular octagon) On 2016-07-30 21:36, James Propp wrote: > I wrote: > > If we're looking at symmetrical solutions, we should probably include even >> values of n back in the game. For instance, when n=8 is there a solution >> with fourfold symmetry? >> > > The answer is YES: e.g., 1,4,3,6,5,8,7,2,1. > > Jim Propp Also, with two mirror axes, the Suicide Schuss triple black diamond: Graphics@Polygon[ReIm /@ (I^({7, 0, 1, 6, 3, 4, 5, 2}/2))] where the middle rhombus is the √2 scaling of a side rhombus. --rwg

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[math-fun] Rectangle-to-Square
by Zak Seidov 31 Jul '16

31 Jul '16

Square-to-Rectangle construction problem is trivial. But what about Rectangle-to-Square construction problem? There are some trivial cases, e.g., if a<b are side lengths such that a*b is a square: (a,b)=(6, 24), (a,b)=(6, 54), etc. But what about general case of any rectangle? zak

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[math-fun] Probability question
by David Wilson 30 Jul '16

30 Jul '16

Let set S contain n objects, each object having equal probability of being selected. At the end of k selections with replacement, what is the expected number of distinct objects that were selected?

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Re: [math-fun] Rectangle-to-Square
by Bill Gosper 30 Jul '16

30 Jul '16

Maybe zak means dissect an arbitrary rectangle into a square, but that's just dissect a rectangle to a square <https://www.google.com/search?client=safari&rls=en&q=dissect+a+rectangle+to…> Or you could do it to right triangles separately in two steps:: gosper.org/A=f(bh).gif (First create some triangle with side √2.) --rwg On 2016-07-30 09:04, James Propp wrote: > If you mean construct-with-compass-and-straightedge, see for instance > > http://planetmath.org/compassandstraightedgeconstructionofgeometricmean > > Jim Propp > > On Saturday, July 30, 2016, Zak Seidov <math-fun(a)mailman.xmission.com> > wrote: > >> Square-to-Rectangle construction problem is trivial. >> But what about Rectangle-to-Square construction problem? >> There are some trivial cases, e.g., if a<b are side lengths >> such that a*b is a square: (a,b)=(6, 24), (a,b)=(6, 54), etc. >> But what about general case of any rectangle? >> zak >>

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Re: [math-fun] True or False, Log[Tan[t + π/4]]
by Dan Asimov 30 Jul '16

30 Jul '16

Good point. —Dan Veit Elser <ve10(a)cornell.edu> wrote: ----- > On Jul 27, 2016, at 6:35 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > > is an odd function of t? —rwg The right question to ask, when it’s multivalued and analytic, is whether the Riemann surface has “odd symmetry”: (z1,z2) \in C^2 is on the surface iff (-z1,-z2) is on the surface. When this is true in an open neighborhood of the origin — as this case which has a regular Taylor series with only odd powers — then it holds for the entire Riemann surface by analytic continuation. -----

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[math-fun] True or False, Log[Tan[t + π/4]]
by Bill Gosper 30 Jul '16

30 Jul '16

is an odd function of t? --rwg

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