November 2017 - mathfun@mailman.xmission.com

[math-fun] 27_4 configuration
by Ed Pegg Jr 17 Nov '17

17 Nov '17

In barycentric coordinates, both lines and points are triples. A point is on a line if point.line = 0. The following set of points AND lines is a configuration, every point is on 4 lines and every line goes through 4 points. This is a 27_4 configuration. {{-2, 3, 1}, {-2, 4, -1}, {-1, -2, 4}, {-1, 0, 2}, {-1, 1, 1}, {-1, 1,2}, {-1, 2, 3}, {-1, 2, 4}, {0, 2, -1}, {0, 2, 1}, {1, -2, 3}, {1, -1, 1}, {1, 0, 2}, {1, 1, -1}, {1, 1, 2}, {1, 2, -1}, {1, 2,1}, {2, -1, 0}, {2, -1, 1}, {2, 1, 0}, {2, 1, 1}, {2, 3, -1}, {2, 4, -1}, {3, -1, 2}, {3, 1, -2}, {4, -1, -2}, {4, -1, 2}} --Ed Pegg Jr

1 0

[math-fun] detailed multi-panel cartoon illustrating a proof in mathematical logic ?
by Thane Plambeck 15 Nov '17

15 Nov '17

i recall looking over some years ago a very nice extended cartoon (it would be more apt to call it a graphical-novel proof, almost) of a theorem in mathematical logic, or maybe category theory. the illustration is not at a high level (ie it's not like, for example LOGICOMIX, but more like XKCD), but the math is real and quite detailed. it was not published book as far as i know. it is also not anything from XKCD. the subject matter was perhaps model theory, but i don't really remember. it features lots of simple characters with thought bubbles that contain embedded other bubbles, where what's in the bubbles is quite helpful for following the logic of the proof. i'd welcome a pointer to it if this rings a bell for anyone. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

3 3

[math-fun] Subject: Re: Draft of my November 2017 blog post
by Bill Gosper 15 Nov '17

15 Nov '17

From: Michael Kleber <michael.kleber(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Cc: James Propp <jamespropp(a)gmail.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >https://xkcd.com/163/ I'm with Knuth (and his home invader) on this one. I like to start my Sums at 0 and my Products at 1: Product[k!^DifferenceDelta[Log[k], {k, 3}], {k, ∞}] == Khinchin^Log[2] Unfortunately, Mathematica starts its Sums at 1 (default) as well. (That guy rappelling through the skylight risked setting off Knuth's pipe organ.) --rwg On Mon, Nov 13, 2017 at 10:58 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > Make sure to run spell check; there are a couple of typos. > > Re do/for loops: > > Mathematicians created the original sin when they used 1-origin array > subscripting. You'd have thought that a thousand years of zero usage would > have taught them something! > > Grace Hopper extended this sin with her 1-origin inclusive do-loops in > Fortran, and we've all paid multiple billions in software debugging costs > ever since. > > The answer is surprisingly simple: > > Always use zero-origin array indexing, and always use half-open interval > do/for loops: > > sum:=0; > for i from 0 upto length(a) do sum:=sum+a(i); > > or > > sum:=0; > N:length(a); > for i in [0,N) do sum:=sum+a(i); > > How so many computer languages could screw this up for so many years is > utterly amazing! > > At 06:37 AM 11/13/2017, James Propp wrote: > >Hi, > > > >I started writing a new draft titled "Impaled on a Fencepost" and would > >love to get people's feedback. I plan on publishing it on the 17th. > > > >Some questions and misgivings that remain in my mind: > > > >Am I missing some fun or important examples of fencepost error? > >Should I cite linguistics literature on the ambiguity of the word > "between" > >(assuming it exists)? > >I feel I'm forgetting some famous movie or TV show featuring the > >fail-to-count-yourself error. > >Is "third, fifths, etc." terminology really called "ordinal" terminology > >for intervals? (I just made that up.) > >Was it really invented during the Renaissance? (I just made that up too.) > >Am I correct in what I write about legal matters? > > > >I forgot to remind everyone that if you think your comment is one that > >might elicit a response from me (e.g., "You're right, my wording was > >confusing; is this better?"), or if you want your contribution to be > >acknowledged, then please sign your response; otherwise I won't know who > >you are. > > > >Please leave your feedback here: > > https://mathenchant.wordpress.com?p=1998&shareadraft=5a0912d2a2e4a > > > >Thanks, > > > >Jim Propp >

1 0

Re: [math-fun] Never so soon
by Dan Asimov 14 Nov '17

14 Nov '17

The entry http://oeis.org/A003504 for OEIS A003504 says: ----- a(0)..a(43) are integral, but from a(44) onwards every term is nonintegral ----- (a sentence that evidently took the collaboration of 3 authors to state correctly in full!). This suggests perhaps that the sequences of numerators and of denominators of the fractional parts might be interesting as well — as far as I could tell, these were not in OEIS. (Aside to Neil: Maybe if room permits, each link to an associated sequence could have next to it a capsule description of itself?) Question: --------- Is there some way to formalize the kind of property that a(n) exhibits and to prove that a(n) (or some other sequence?) is the "simplest" one with that property? —Dan Tom Rockicki wrote: ----- > I'm curious to find out what the fractional part of the number will turn out to be. I think it'll be some integer divided by 43. As shown by Guy in the paper (which is scanned with annotations here: http://oeis.org/A005165/a005165.pdf) you'll see that the fractional part is simply 24/43 (see page 709, near his written A3504.) On Tue, Nov 14, 2017 at 7:47 AM, Hans Havermann <gladhobo(a)bell.net> wrote: > Yesterday, Cliff Pickover's twitter feed presented a bit from Pickover's > 2005 "A Passion for Mathematics" which references Guy's 1994 "Unsolved > Problems in Number Theory" (2nd ed.) E15, a recursion of Göbel, wherein is > stated that x(43) of the sequence is not an integer. The sequence is > A003504: > > http://oeis.org/A003504 > > There's a different offset in the OEIS version, so A003504(44) is now the > first one that is not an integer. Pickover in his book felt the need to add > something to the problem so, noting that A003504(44) = > 5.4093*10^178485291567, he stated that this number "is so large that > humanity will *never* be able to compute all of its digits". > > I had a go on my four-year-old Mac Pro with 64 GB RAM and was only able to > compute A003504(42) with its 44621322894 decimal digits. That suggests that > when the next iteration of the Mac Pro, with 256 GB RAM, comes out in 2018 > it should be able to calculate the number. But I know that there are > personal computer setups out there right now that enjoy 256 GB RAM, so I > emailed Cliff with a "never is now". :) > > I'm curious to find out what the fractional part of the number will turn > out to be. I think it'll be some integer divided by 43. -----

2 1

[math-fun] 30 decimal places obtained at each iteration.
by François Mendzina Essomba 14 Nov '17

14 Nov '17

30 decimal places obtained at each iteration. a(n)=(2^(n+1)*u(n)*sqrt(1-u(n)^2)*(136875399077482387003872084531664125952*u(n)^98+138272086823170982789625881312599474176*u(n)^96+139712421060912347193684484242939052032*u(n)^94 +141198723412624180674468361734885212160*u(n)^92+142733492145370095681799539579829616640*u(n)^90+144319419835874207856041756686272167936*u(n)^88 +145959413243100051127133140284979806208*u(n)^86+147656615722670981954192827962712129536*u(n)^84+149414432576512303167933218771792035840*u(n)^82 +151236559803055136133395819000716328960*u(n)^80+153127016800593325335063266738225283072*u(n)^78+155090183682652214121410231696407658496*u(n)^76 +157130843994266059044060366323991969792*u(n)^74+159254233777972357139250371274316185600*u(n)^72+161466098135999750988406626430903910400*u(n)^70 +163772756680799747431098149665631109120*u(n)^68+166181179573164449599202534219537448960*u(n)^66+168699076233363910956766208980439531520*u(n)^64 +171334999299510222065465680995758899200*u(n)^62+174098467030147483711682869398916300800*u(n)^60+177000108147316608440210917222231572480*u(n)^58 +180051834149856549965042139932959703040*u(n)^56+183267045473961131214417892431762554880*u(n)^54+186660879649404855866536742291610009600*u(n)^52 +190250511950354949248585525797217894400*u(n)^50+194055522189362048233557236313162252288*u(n)^48+198098345568307090905089678736353132544*u(n)^46 +202404831341531158098678584795839070208*u(n)^44+207004941144747775328194007177562685440*u(n)^42+211933630219622722359817674015123701760*u(n)^40 +217231970975113290418813115865501794304*u(n)^38+222948601790247850692992408388278157312*u(n)^36+229141618506643624323353308621285883904*u(n)^34 +235881077874486083862275464757206056960*u(n)^32+243252361558063773982971573030868746240*u(n)^30+251360773609999233115737292131897704448*u(n)^28 +260337944096070634298442195422322622464*u(n)^26+270350941945919504848382279861642723328*u(n)^24+281615564526999484217064874855877836800*u(n)^22 +294416272005499460772386005531145011200*u(n)^20+309137085605774433811005305807702261760*u(n)^18+326311368139428569022727822797019054080*u(n)^16 +346705828648142854586648311721832744960*u(n)^14+371470530694438772771408905416249369600*u(n)^12+402426408252308670502359647534270150400*u(n)^10 +442669049077539537552595612287697165440*u(n)^8+498002680212231979746670063823659311120*u(n)^6+581003126914270643037781741127602529640*u(n)^4 +726253908642838303797227176409503162050*u(n)^2+1089380862964257455695840764614254743075))/(363126954321419151898613588204751581025) ; u(n)=sqrt(2-2*w(n))/2 ; w(n+1)=sqrt(2+2*w(n))/2 ; w(0)=1/2 ; n --- infinity ; a(n)--- Pi ;

1 0

Re: [math-fun] Miscounting in the Cheddar Gorge ...
by Keith F. Lynch 14 Nov '17

14 Nov '17

Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > As a rather nerdy schoolboy, I read somewhere an apocryphal quote by > an Alaskan old-timer that "in Winter, the temperature stays forty > below". For many years I wondered now and again whether this was in > degrees Centigrade or Fahrenheit. He may have meant only one or the other, whether or not he knew that they were the same. Which reminds me of the question about how many numeric dates are ambiguous, given that they can be written with the month either first or second. For instance 7/4 can mean July 4th or the 7th of April, but there's no ambiguity about 11/13, since there's no 13th month. You can argue for either 144 or 132, depending on whether you consider a date such as 9/9 as ambiguous. Does it mean September 9th or the 9th of September? They're the same date, but it remains ambiguous which 9 in 9/9 refers to the month.

1 0

Re: [math-fun] Distance across integer coordinate planes
by Dan Asimov 14 Nov '17

14 Nov '17

Since that turned out to be pretty easy, here's a related problem: ----- Consider the 4 families of parallel planes given by ± x ± y ± z = K for all integers K. Let the union of all such planes be denoted by V. For any integer point P = (K,L,M), how long is the shortest path connecting P to (0,0,0) that lies entirely in V ??? —Dan Allan Wechsler wrote: ----- You have a box bounded by the planes x=0, x=K, y=0, y=L, z=0, z=M. My intuition is that confining the path to the surface of that box doesn't cost you anything. If that intuition is correct, the answer is sqrt(min((K+L)^2 + M^2, (K+M)^2 + L^2, (L+M)^2 + K^2)). On Mon, Nov 13, 2017 at 7:57 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: Let W denote the union of all coordinate planes given by x = K or y = K or z = K for K an integer. I.e., W = Z x R x R u R x Z x R u R x R x Z with Z = integers, R = reals. Question: --------- Let P, Q belong to Z^3. What is the length of the shortest path connecting P to Q that lies entirely in W ??? * * * WLOG assume P = (K,L,M) and Q = (0,0,0). The answer is a function of form f(K,L,M). What is f(K,L,M) ??? I'm not sure if this is easy or hard. -----

1 0

[math-fun] Distance across integer coordinate planes
by Dan Asimov 14 Nov '17

14 Nov '17

Let W denote the union of all coordinate planes given by x = K or y = K or z = K for K an integer. I.e., W = Z x R x R u R x Z x R u R x R x Z with Z = integers, R = reals. Question: --------- Let P, Q belong to Z^3. What is the length of the shortest path connecting P to Q that lies entirely in W ??? * * * WLOG assume P = (K,L,M) and Q = (0,0,0). The answer is a function of form f(K,L,M). What is f(K,L,M) ??? I'm not sure if this is easy or hard. —Dan

3 2

Re: [math-fun] Draft of my November 2017 blog post
by Henry Baker 13 Nov '17

13 Nov '17

Make sure to run spell check; there are a couple of typos. Re do/for loops: Mathematicians created the original sin when they used 1-origin array subscripting. You'd have thought that a thousand years of zero usage would have taught them something! Grace Hopper extended this sin with her 1-origin inclusive do-loops in Fortran, and we've all paid multiple billions in software debugging costs ever since. The answer is surprisingly simple: Always use zero-origin array indexing, and always use half-open interval do/for loops: sum:=0; for i from 0 upto length(a) do sum:=sum+a(i); or sum:=0; N:length(a); for i in [0,N) do sum:=sum+a(i); How so many computer languages could screw this up for so many years is utterly amazing! At 06:37 AM 11/13/2017, James Propp wrote: >Hi, > >I started writing a new draft titled "Impaled on a Fencepost" and would >love to get people's feedback. I plan on publishing it on the 17th. > >Some questions and misgivings that remain in my mind: > >Am I missing some fun or important examples of fencepost error? >Should I cite linguistics literature on the ambiguity of the word "between" >(assuming it exists)? >I feel I'm forgetting some famous movie or TV show featuring the >fail-to-count-yourself error. >Is "third, fifths, etc." terminology really called "ordinal" terminology >for intervals? (I just made that up.) >Was it really invented during the Renaissance? (I just made that up too.) >Am I correct in what I write about legal matters? > >I forgot to remind everyone that if you think your comment is one that >might elicit a response from me (e.g., "You're right, my wording was >confusing; is this better?"), or if you want your contribution to be >acknowledged, then please sign your response; otherwise I won't know who >you are. > >Please leave your feedback here: > https://mathenchant.wordpress.com?p=1998&shareadraft=5a0912d2a2e4a > >Thanks, > >Jim Propp

3 2

[math-fun] Miscounting errors
by James Propp 13 Nov '17

13 Nov '17

Anyone have a favorite puzzle in which miscounting plays a role? My favorite is the classic bookworm puzzle (see https://math.stackexchange.com/questions/1271651/how-is-this-true-bookworm-… ). Another example: A man was born in 50 BC and died on the same day in 50 AD. How old was he when he died? I'm especially interested in puzzles that lend themselves to solvers committing fencepost errors, and off-by-one errors more generally. Jim Propp

6 5