April 2018 - mathfun@mailman.xmission.com

[math-fun] “Random” cards
by James Propp 13 Apr '18

13 Apr '18

If you ask people to name a random card in a standard deck, what regularities govern their answers? (Is this something someone has looked at?) Cf. the observation that when people are asked to pick a number between 1 and 20, the most common answer is 17. Jim Propp

9 10

[math-fun] Radial disk-dissection
by James Propp 13 Apr '18

13 Apr '18

If you dissect a unit disk radially into a large number of equal wedges, it’s well known that you can reassemble them to form a shape that in the limit converges to a 1-by-pi rectangle. What other limiting shapes can we form in the limit from patterns that join the wedges edge-to-edge? I’m guessing that you get a class of curvy pseudoquadrilaterals ABCD where AB is a unit segment, CD is a unit segment, and the curves BC and AD satisfy some sort of curvature-compatibility condition. Jim Propp

1 0

Re: [math-fun] "Random" cards
by Henry Baker 13 Apr '18

13 Apr '18

How big a % of the population has any clue what a "random" card means? It only took mathematicians a few *hundred* years to figure this out, and perhaps they still aren't done yet. I'm not blaming the mathematicians; these are very, very subtle and tricky notions, so it doesn't surprise me that Ivy Leaguers -- particularly the ones in Washington, DC -- don't really understand these things either. Perhaps the minimum wage folk who purchase lottery tickets have a better idea that most about what "random" means. At 06:06 PM 4/12/2018, James Propp wrote: >If you ask people to name a random card in a standard deck, what >regularities govern their answers?

3 2

[math-fun] The Moor of Venice
by françois mendzina essomba2 12 Apr '18

12 Apr '18

Did you read Othello my dear ?? a recurrence for functions a(n+1)=2^(n+3/2)*sqrt(1-sqrt(1-2^(-2*n-2)*a(n)^2)) ; for : a(0)=sqrt(2-2/sqrt(1+x^2)) ; n -------- infinity ; a(n) ------- atan(x) ; for : a(0)=sqrt(2-2*x) ; n -------- infinity ; a(n) ------- acos(x) ; for : a(0)=sqrt(2-2*sqrt(1-x^2)) ; n -------- infinity ; a(n) ------- asin(x) ; for : a(0)=sqrt(2-2*x/sqrt(1+x^2)) ; n -------- infinity ; a(n) ------- acot(x) ; The results are in absolute values FME...

1 0

[math-fun] Generalized stable fummin problem
by Dan Asimov 12 Apr '18

12 Apr '18

I found Jim Propp's idea of generalizing a problem to irrational numbers very interesting. So can this be done for the fummin problem? Hmm. Suppose we have N unit-circumference circles that we want to distribute evenly among K recipients. Then the "(N,K) fummin problem" is to find the smallest number f(N,K) of pieces that can be rearranged to give each recipient a total length of N/K. The *stable* fummin problem tries to find the limit s(N,K) = lim (1/m)f(mN,mK), m—>oo if it exists. (Question: Does s(N,K) <> f(N,K) for some (N,K) ?) Clearly s(N,K) depends only on the ratio N/K so by abuse of notation we define s(N/K) = s(N,K). Let alpha be a positive real number. Then the "generalized stable fummin problem" is to find the limit g(alpha) = lim s(N_j / K_j), j—>oo where N_j / K_j is a sequence of rational numbers converging to alpha, *if* the limit exists and is independent of the sequence N_j / K_j approaching alpha. Question: Is g(alpha) defined for any (all?) irrational numbers alpha > 0 ??? —Dan

1 0

[math-fun] Is mathematical truth "real" ?
by Dan Asimov 12 Apr '18

12 Apr '18

If this common discussion subject among mathematicians has come up in math-fun previously, it must have been a while ago: Is mathematical truth real? I was reminded of this question when I happened to sit next to an esteemed molecular biologist who has strong opinions on the matter: He thinks math is "all in the brain" of humans who think about it. I could not convince him that math has any kind of independent existence — though I certainly believe it myself. What do other think about this? I would say that mathematical truth is *at least* as real of a thing as physical truth. —Dan

9 10

Re: [math-fun] Who's going to G4G13?
by Keith F. Lynch 11 Apr '18

11 Apr '18

Not me, though I'd like to. Please have the people in charge ask the people who run the annual World Science Fiction Convention on how the latter manage to put on an event that lasts twice as long for half the price with no membership limit.

4 3

Re: [math-fun] Miss those IBM/DEC blinking lights?
by Henry Baker 11 Apr '18

11 Apr '18

I'm surprised that auto LED headlights don't already broadcast their vehicle VIN# (maybe they do?). Apparently, one of the countries in the Middle East is now requiring some sort of RFID IFF for automobiles. Sure makes the CIA drone program a lot more accurate... I've heard that Disney is starting to replace some of its nighttime *fireworks* displays with *LED drone fleets* displays. I would imagine that some of these temporal streaks could be useful for these drone displays. At the last several CES's (Consumer Electronics Shows), I talked with vendors making bicycle LED displays, and none of them were very imaginative. I did see a T-shirt with a bit-mapped display, but it *wasn't programmable* !! What weren't they thinking? How come I still can't buy a T-shirt with a bitmapped display on it? -- it might be great for political protests, etc. It might be fun to hack the news cameras by synchronizing the T-shirt display with a GPS-derived 60Hz signal; with proper synchronization, it might be possible to have the TV news cameras pick up completely different content from that perceived by human observers! I seem to recall that someone at MIT had developed an LED display for bicycle spokes that was time-multiplexed in such a way as to provide a full circular bit-mapped display -- kinda like those WWII radar displays. At 01:28 AM 4/10/2018, Bill Gosper wrote: >It should also be possible to Morse-code messages by rapidly blinking LED auto taillights to leave dotted and dashed afterstreaks, if you know how to look.

4 3

[math-fun] The mango lassi problem (and meta-problem)
by James Propp 11 Apr '18

11 Apr '18

I had dinner last night with Alan Frank (the inventor of the muffin problem that was discussed on math-fun a few years back and which Bill Gasarch will be talking about at the MIT Combinatorics Seminar tomorrow), and we encountered a problem about sharing that might turn out to be a sequel to the muffin problem, except that I don't quite know what the right problem is. (I solved the special case I needed to solve, but I'm not sure what the right general setting is.) Dining with me and Alan were three kids. I ordered two mango lassis to split between myself and the three kids, along with two extra glasses, so that each of the four lassi-drinkers would get half a serving. The two glasses containing (identical amounts of) mango lassi were the same shape as one another, and the two empty glasses were the same shape as one another, but the empty glasses were NOT the same shape as the other two glasses. Not a problem! I took a full glass of lassi and divided it evenly between the two initially empty glasses, and then I took the remaining full glass and divided the lassi it contained evenly between it and the glass I'd drained in the first step. But what if the two empty glasses had NOT been the same shape as each other? (We may assume that each is big enough to hold a full serving of mango lassi.) PROBLEM: Show that even if the two empty glasses were different shapes from one another (and from the identical full glasses), one can still divide the lassi into four equal portions. The only allowed operation is to equalize the amount of lassi in two glasses of the same shape by equalizing the height of the lassi. (One might quibble that in practice you can only do this approximately, but for purposes of this puzzle, ignore that nicety.) META-PROBLEM: What's the best way to alter and/or extend this problem, where "best" means "most fun"? Jim Propp Jim Propp

3 7

[math-fun] Viete for inverse sinusoidal functions
by françois mendzina essomba2 10 Apr '18

10 Apr '18

(2/sqrt(2*x/(sqrt(x^2+1))+2))*(2/sqrt(sqrt(2*x/(sqrt(x^2+1))+2)+2))*(2/sqrt(sqrt(sqrt(2*x/(sqrt(x^2+1))+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2+1))+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2+1))+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2+1))+2)+2)+2)+2)+2)+2))*... = sqrt(x^2+1)*acot(x); (2/sqrt(2/(sqrt(-x^2+1))+2))*(2/sqrt(sqrt(2/(sqrt(-x^2+1))+2)+2))*(2/sqrt(sqrt(sqrt(2/(sqrt(-x^2+1))+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2/(sqrt(-x^2+1))+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2/(sqrt(-x^2+1))+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2/(sqrt(-x^2+1))+2)+2)+2)+2)+2)+2))*... = sqrt(1/x^2-1)*atanh(x); (2/sqrt(2*x/(sqrt(x^2-1))+2))*(2/sqrt(sqrt(2*x/(sqrt(x^2-1))+2)+2))*(2/sqrt(sqrt(sqrt(2*x/(sqrt(x^2-1))+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2-1))+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2-1))+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*x/(sqrt(x^2-1))+2)+2)+2)+2)+2)+2))*... = sqrt(x^2-1)*acoth(x); (2/sqrt(2*x+2))*(2/sqrt(sqrt(2*x+2)+2))*(2/sqrt(sqrt(sqrt(2*x+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2)+2)+2))*... = acosh(x)/sqrt(x^2+1); (2/sqrt(2*sqrt(x^2+1)+2))*(2/sqrt(sqrt(2*sqrt(x^2+1)+2)+2))*(2/sqrt(sqrt(sqrt(2*sqrt(x^2+1)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*sqrt(x^2+1)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*sqrt(x^2+1)+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*sqrt(x^2+1)+2)+2)+2)+2)+2)+2))*... = asinh(x)/x; (2/sqrt(2*sqrt(-x^2+1)+2))*(2/sqrt(sqrt(2*sqrt(-x^2+1)+2)+2))*(2/sqrt(sqrt(sqrt(2*sqrt(-x^2+1)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*sqrt(-x^2+1)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*sqrt(-x^2+1)+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*sqrt(-x^2+1)+2)+2)+2)+2)+2)+2))*... = asin(x)/x; (2/sqrt(2*x+2))*(2/sqrt(sqrt(2*x+2)+2))*(2/sqrt(sqrt(sqrt(2*x+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2)+2))*(2/sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2*x+2)+2)+2)+2)+2)+2))*... = acos(x)/sqrt(1-x^2);

1 0