October 2006 - mathfun@mailman.xmission.com

[math-fun] continued subscripts
by Marc LeBrun 10 Nov '06

10 Nov '06

Recent correspondence has inspired me that, analogous to stacking superscripts for iterated exponentiation, we may stack the subscripts that are vulgarly used as "base" notation. For example, we may apprehend, among 239's many sterling properties, the following gratuity: 10 11 12 13 14 15 16 = 239. 17 18 19 20 21 22 23 That is, working up from the bottom: ... ... ... 21 = 21 = 97 and so on = 239. 22 48 23 Question: If stacking superscripts produces a "tower", what are stacked subscripts? A "dungeon"? The sequence following this incrementing pattern of subscripts, namely, 10, 10, 10, 10, ... 11 11 11 12 12 13 has (if I've computed them correctly) the values 10 11 13 16 20 25 31 38 46 55 65 87 135 239 463 943 1967 4143 8751 18479 38959 103471 306223 942127 2932783 9153583 28562479 89028655 277145647 861652015 2675637295 10173443119 41132125231 168836688943 695134284847 2862242032687 11778342480943 48433422101551 199016451894319 817199416306735 3353334654921775 16351027752823855 82924577766468655 423717750455364655 2167310727003204655 11083411175259204655 56654591244123204655 289463880726363204655 1478277273827163204655 7546178967779163204655 38504861079779163204655 227971995605219163204655 1387065053878499163204655 8475364910242019163204655 51807613088766179163204655 ... (which I will submit to the OEIS as soon as I can craft a concise explanation!). Puzzle: the least-significant digits seem to be stabilizing ... why or why not? Of course we can write the numerals in the stack in other bases, such as binary: 10, 10, 10, 10, ... 11 11 11 100 100 101 and these have as their first few values 2 3 5 26 1370 9840770 851566070376026 ... but it grows rapidly--the next two terms have ~144 and ~450 digits respectively (and seem beginning to show a distinct tendency to end with the digits ...26 ?!) Question: What is a good approximate formula for a(n)? And of course other continued subscript patterns besides simply incrementing the terms are possible.

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Re: [math-fun] Games of Ramsey and van der Waerden
by Daniel Asimov 30 Oct '06

30 Oct '06

David asks: << Can anybody tell me who is the winner of the following two games. Game of Ramsey. Two players alternately pick edges from a complete graph on 6 vertices until one of them has collected three edges that form a triangle at which point his Opponent is the winner. Game of van der Waerden. Two players alternately pick numbers from the set {1,2, . .,9} until one of them has collected three numbers which form an arithmetic progression at which point his Opponent is the winner. The point is, of course, that these games cannot end in a draw. >> Darned if I know, but very interesting questions! I'd like to add (two variants of) another game, which Martin Gardner and I (and probably innumerable others) invented independently, or at least close variants of the same idea: Who wins in the Game of Squares: Players take turns placing a marker of their own color (say white or black) on the vertices of an 8x8 square grid until there is some square (tilted or not) with all four vertices having the same-colored markers on them. The player whose color it is loses. Same question for the slightly more symmetrical Game of Toral Squares: Same game but on the 64 vertices of GxG, where G is a regular octagon. (Microchallenge: How many distinct squares are there in the GxG case, where a square is defined by its set of vertices? Better, generalize to the case of G = the regular N-gon) --Dan

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OOPS -- sorry Re: [math-fun] Games of Ramsey and van der Waerden
by Daniel Asimov 30 Oct '06

30 Oct '06

Gareth, Ohhh, now *I* see what you mean by the metric inherited from R^2, which of course *does* depend on the embedding. I'm so used to thinking about Riemannian metrics, that this is wha I was referring to on simple closed curves (but without saying so). (Sorry about that. At least I had the wisdom not to post it to the group.) But all's well that ends well, right? Regards, Dan << Ohhh, I see. I entirely misunderstood. I thought you meant the metric induced by the obvious embedding of G in R^2. With the clarification you've now provided, I agree that that's a fine definition, not least because it's equivalent to the one I prefer :-). >>

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Re: [math-fun] Games of Ramsey and van der Waerden
by Daniel Asimov 30 Oct '06

30 Oct '06

Gareth wrote: << On Monday 30 October 2006 00:57, Daniel Asimov wrote: > A square is defined here as four *distinct* points of GxG -- P,Q,R,S -- > such that the segments PQ, QR, RS, ST are of equal length, and the > angles PQR, QRS, RSP, SQP are right. A square is determined by its > set of vertices. > > Oh -- and the metric used on GxG is just the cartesian product of > that on G with itself, where G is a regular octagonal curve. OK. Allow me to suggest that that's less natural than the different definition I proposed, because it seems freaky to consider {a1,c1,e1,g1} a square. But it's your game, so you get to choose the rules :-). >> If you mean (0,0),(0,2),(0,4),(0,6), that's *not* a square -- all angles are straight, not right. A regular octagonal curve has the same metric as a circle (with 8 equally spaced points on it). The 64 vertices derive their metric from the metric on S^1 x S^1 = R^2 / Z^2 (as metric spaces). The distance between two points P,Q of R^2 / Z^2 is the length of the shortest line segment in R^2 whose endpoints project to P,Q. PQR is a right angle when the shortest segments PQ,QR make a right angle, or equivalently when they have lifts PQ^, QR^ in perpendicular directions R^2. Sorry to have been unclear. (Or am I missing something?) --Dan

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Re: [math-fun] Games of Ramsey and van der Waerden
by Daniel Asimov 30 Oct '06

30 Oct '06

Gareth asks: << [I wrote]: > Game of Squares: Players take turns placing a marker of their > own color (say white or black) on the vertices of an 8x8 square > grid until there is some square (tilted or not) with all four vertices > having the same-colored markers on them. The player whose color it is > loses. ... > Game of Toral Squares: Same game but on the 64 vertices of GxG, > where G is a regular octagon. How are you defining squares formed from the vertices of GxG? >> A square is defined here as four *distinct* points of GxG -- P,Q,R,S -- such that the segments PQ, QR, RS, ST are of equal length, and the angles PQR, QRS, RSP, SQP are right. A square is determined by its set of vertices. Oh -- and the metric used on GxG is just the cartesian product of that on G with itself, where G is a regular octagonal curve. --Dan

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[math-fun] Games of Ramsey and van der Waerden
by David Gale 29 Oct '06

29 Oct '06

Can anybody tell me who is the winner of the following two games. Game of Ramsey. Two players alternately pick edges from a complete graph on 6 vertices until one of them has collected three edges that form a triangle at which point his Opponent is the winner. Game of van der Waerden. Two players alternately pick numbers from the set {1,2, . .,9} until one of them has collected three numbers which form an arithmetic progression at which point his Opponent is the winner. The point is, of course, that these games cannot end in a draw. David

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[math-fun] minimax polynomials
by J.P. Grossman 29 Oct '06

29 Oct '06

Does anyone know of any references for finding the minimiax polynomial approximation to x^N on some interval that has degree k < N? Thanks, J.P. Grossman

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[math-fun] Yet another math-based drama
by Daniel Asimov 26 Oct '06

26 Oct '06

A new play about Hardy and Ramanujan just opened off-Broadway: < http://theater2.nytimes.com/2006/10/21/theater/reviews/21man.html >. --Dan

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[math-fun] Dual curve sequence
by Fred lunnon 23 Oct '06

23 Oct '06

The plane curve defined as the locus of the (homogeneous) point (X,Y,Z) satisfying the equation X^n + Y^n + Z^n = 0 has a dual form as the envelope of (tangent) line A x + B y + C z = 0, where the (A,B,C) satisfy a polynomial equation of degree m = n(n-1) --- more specifically, a polynomial of degree n-1 in A^n, B^n, C^n. Example --- n = 4, point poly = X^4 + Y^4 + Z^4, dual poly = A^12 + 3*A^8*B^4 + 3*A^8*C^4 + 3*A^4*B^8 - 21*A^4*B^4*C^4 + 3*A^4*C^8 + B^12 + 3*B^8*C^4 + 3*B^4*C^8 + C^12 Question --- why does m = n(n-1) ? The coefficients (1 and 3 above) of A^(k+1)n B^(n-k) appear to be binomial coeffcients; but instead of the multinomial 6 expected, we find 21 above. So define S_n = - coefficient of A^n(n-3) B^n C^n in the dual polynomial. Magma (1000 times faster than Maple by the way!) finds for n = 0..12 S_n = 0 0 0 0 2 21 605 2736 11963 51416 218709 923680 3879755 ... Question --- what is this sequence? (OEIS doesn't recognise it) Fred Lunnon

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[math-fun] addition-multiplication chains
by Schroeppel, Richard 20 Oct '06

20 Oct '06

Awfully quiet around here lately. Everyone busy with their Halloween preparations? I've been playing with Addition-Multiplication chains (AMchains). These are like addition chains, but you can use multiplication too. Each chain entry is the sum or product of two earlier entries. You begin with 1; squaring is allowed. The goal is to come up with the minimal chain for a number. I've decided for coding purposes that the entries must always increase, but it's not a part of the definition. Some examples: 3141: 1 2 4 5 16 25 625 3125 3141 (9 links) 31415: 1 2 3 5 10 100 103 300 305 31415 (10 links) 314159: 1 2 4 8 64 68 69 137 4416 4553 314157 314159 (12 links) It seems like the number of links should be somewhere between O(logN) and O(logN/loglogN). The smallest numbers that require each chain size are 1 1 1 2 2 1 2 3 3 1 2 3 5 4 1 2 3 5 7 5 1 2 3 4 7 13 6 1 2 3 4 9 13 23 7 1 2 3 4 5 20 23 59 8 1 2 3 4 7 8 56 59 211 9 1 2 3 4 7 14 15 210 211 619 10 1 2 3 4 7 8 11 56 616 619 4282 11 1 2 3 4 7 8 56 57 65 4225 4282 Chains seem to have a lot of equivalents, available by playing around with the terms. For example, the sequence X, X^2, X^2 + X can also be done as X, X+1, X^2+X. The table of minimal chain lengths is pretty boring. 0 01233445445556654556566755656667566656666767767766 50 67776777786777566767786776777865666777776777886776 100 67777778677778787788767776776778788778887887677778 ... 1000 78888889888989899999899967787889788989997889899989 1050 8998997889899989999A998999999989989999778888998889 1100 898988889998999A899988899788898999889999899999999A The short-chain value 2^10 = 1024 causes only a minor disturbance in the pattern. The shortest chains for 2^K always seem to be 2^(minimal addition chain). Rich

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