February 2017 - mathfun@mailman.xmission.com

[math-fun] Fwd: A sandpile game
by Bill Gosper 04 Feb '17

04 Feb '17

---------- Forwarded message ---------- From: Gary Snethen <gsnethen(a)gmail.com> Date: Fri, Feb 3, 2017 at 5:39 PM Subject: Re: [math-fun] A sandpile game To: Bill Gosper <billgosper(a)gmail.com> "Ripples" ---Gary On Fri, Feb 3, 2017 at 2:46 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > Date: 2017-02-03 09:22 > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Reply-To: math-fun <math-fun(a)mailman.xmission.com> > > I’ve devised a combinatorial game based on the theory of the abelian > sandpile model and I’m trying to come up with a good name for it. > > The game is played using identical tokens on a board consisting of nine > consecutive boxes, each of which is capable of holding one or two tokens. > Players take turns. When it’s your turn to move, you add a token to any of > the nine boxes; if the box already contains a token, then the box “fires” > (by sending one token to the left and one token to the right), and if that > leads one or more boxes to contain more than one token, then those boxes > fire (by sending one token to the left and one token to the right), and so > on; when the leftmost or rightmost box fires, the token that gets fired off > the board disappears from the game. Eventually the board stabilizes so that > each box contains at most one token. > > Example: If the state of the board is > > 1 1 1 0 1 0 0 1 0 > > and a player adds a token to the second square from the left, the state of > the board evolves as follows: > > 1 2 1 0 1 0 0 1 0 > 2 0 2 0 1 0 0 1 0 > 0 2 0 1 1 0 0 1 0 > 1 0 1 1 1 0 0 1 0 > > The goal of the game is to be the player who achieves the configuration > > 1 1 1 1 1 1 1 1 1 > > (I’ll post a video sometime soon that will make the nature of game-play > much clearer than the above, but I’d like to have a good name for the game > first.) > > There’s a very nice (but non-obvious) numerical way to look at the game > that makes it quite easy to recognize, from any given starting position, > whether it’s a win for the first player or the second player, and the > winning strategy is also quite simple. > > I’m pretty sure this is new. Can anyone think of a good name for it? > > Please give me feedback on this game. (It might be more apt to call it a > puzzle in which the goal is to figure out how to play the game well.) > > Jim Propp >

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Re: [math-fun] Gamma function question
by Don Reble 03 Feb '17

03 Feb '17

> y = |gamma(x)| > for negative x has one tangency in each interval (-n, -n+1) (n > 0) > with the graph of some nice convex function > y = g(x) > ... Sort of an "envelope". > If so, is there a unique "nicest" g that works here? Is g = pi / gamma(1-x) nice enough? -- Don Reble djr(a)nk.ca

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[math-fun] Logic question
by David Wilson 03 Feb '17

03 Feb '17

Can a reductio ad adsurdum argument be rearranged into a direct argument that does not require a supposition? In short, are suppositions necessary for logical arguments?

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[math-fun] Logic question
by Don Reble 03 Feb '17

03 Feb '17

> ... are suppositions necessary for logical arguments? Thomason, "Symbolic Logic" (Collier-Macmillan, 1970), shows a procedure for removing a supposition from a proof (p68-69). That mechanical process (at least) triples the length of the proof-part within the supposition. I suppose something similar applies to any proof system. -- Don Reble djr(a)nk.ca

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Re: [math-fun] Torus tiling problem
by Dan Asimov 02 Feb '17

02 Feb '17

This problem was inspired by having read some cool theorems about tiling higher-dimensional things with simplices (n-dimensional polytopes that are isometric to the convex hull of n+1 points in general position in n-space — the natural generalization of a triangle). Definitions: * An n-simplex is "acute" if all of its dihedral angles (there are n(n+1)/2 of them, one for every pair of (n-1)-dimensional faces) is acute. * A "flat" n-manifold is one such that every point has a neighborhood that is isometric with some open ball in n-dimensional Euclidean space. * An "acute triangulation" of a flat manifold X is a way of expressing X as the union of acute simplices with disjoint interiors such that any (n-1)-face of any simplex belongs to exactly two simplices. * The cubical n-torus is R^n / Z^n, the result of identifying opposite (n-1)-faces of the unit n-cube. Theorem: -------- There is an acute triangulation of the 3-cube having 1370 tetrahedra. (This is apparently the fewest known.) Theorem: -------- There exists no acute triangulation of n-space for n >= 5. Theorem: -------- There exists no acute triangulation of the cubical n-torus for n >= 4. (It is unknown whether there exists any acute triangulation of 4-space.) Theorem: -------- If there exists an acute triangulation of 4-space, then the supremum of its set of dihedral angles must be pi/2. –Dan > On Jan 31, 2017, at 6:16 AM, Veit Elser <ve10(a)cornell.edu> wrote: > > By Euler the average degree of a vertex is 6, and the number of triangles is twice the number of vertices. I can construct a tiling from 3 vertices, for 6 triangles (first slice Z^2 on square diagonals to make strips, then subdivide strips). Ruling out a tiling with just one vertex looks doable, but the two vertex case might be tedious … > > Nice problem. > > -Veit > >> On Jan 30, 2017, at 7:43 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> >> ----- >> QUESTION: Given the square torus T^2, what is the smallest number of triangles in a tiling of the torus by triangles such that they are all acute? >> >> I need to define a few terms for this particular question: >> >> * The "square torus" T^2 is the result of identifying opposite edges of a square. (Equivalently, it is the metric space obtained from the quotient of the plane R^2 by its integer subgroup Z^2.) >> >> * A "tiling of the torus by triangles", for the purposes of this question, is any union of triangles that equals the torus, such that >> >> a) Any two triangles have disjoint interiors, >> >> and >> AAAA >> b) No triangle's vertex lies on the interior of any triangle's edge. >> ----- >> >> It's well known that the square can be tiled in this sense with 8 acute triangles and no fewer. The standard example (https://www.ics.uci.edu/~eppstein/junkyard/acute-square/8-square.gif) gives, by identifying opposite edges of the square, a tiling of the square torus using 8 acute triangles. >> >> Note: The most general flat torus is obtained by identifying opposite edges of a parallelogram. Clearly, any non-rectangular flat torus can be tiled in the sense of this question using only 2 acute triangles. >> >> Can the upper bound of 8 for the square torus be improved on? And what about a non-square rectangular torus? >> >> —Dan >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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[math-fun] Algebra question
by David Wilson 01 Feb '17

01 Feb '17

Consider the set S of sequences where the nth element is a residue of the nth prime. Natural elementwise addition and multiplication operations can be defined on S. With respect to these operations, S has unique additive and multiplicative identities (0 and 1), as well as additive and multiplicative inverses. S also has a subset isomorphic to the integer, where integer k corresponds to the sequence whose nth element is k modulo the nth prime. Not being a number theorist, I was wondering about the algebraic structure of this set.

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