July 2017 - mathfun@mailman.xmission.com

[math-fun] Minimal Squaring Conjecture counterexample found
by Ed Pegg Jr 09 Jul '17

09 Jul '17

If an m×n rectangle is minimally divisible into k integer-sided squares, it it ever the case that a 2m×2n rectangle can be divided into fewer squares? https://mathoverflow.net/questions/116382/tiling-a-rectangle-with-the-small… The (7125×7081) rectangle seems to need 21 squares. The 2(7125×7081) rectangle only needs 20 squares. Pictures at the link. --Ed Pegg Jr

1 0

[math-fun] I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial
by Bill Gosper 09 Jul '17

09 Jul '17

On 2017-07-08 11:15, Mike Speciner wrote: > You're assuming a flat earth? > Isn't it? --rwg > On 08-Jul-17 13:38, Bill Gosper wrote: >> Oops, I must have been unclear. Baseball home run trajectories are usually >> longest with well *below* 45º elevation and lots of backspin. >> But I was thinking of cannonballs. In a vacuum, 45º is best. With air >> resistance, >> best is somewhat lower. But really long range artillery goes farther >> *above* 45º by >> getting a few miles up into thinner air. (As North Korea is threatening to >> demonstrate.) >> I was speculating that there might be a case where 45º is a local pessimum. >> --rwg >> >> On 2017-07-08 07:15, Fred Lunnon wrote: >>> This looks a good place to start answering my own question --- >>> >>> >> https://physics.stackexchange.com/questions/200097/flat-throw-vs-45-degree-… >>> WFL >>> >>> >>> >>> On 7/8/17, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >>>> Which reminds me of a recent RWG claim that a baseball travels further >>>> along a horizontal trajectory under gravity at some elevation inceeding >>>> pi/4 . >>>> >>>> How does this come about exactly --- is upward pitching spin involved? >>>> WFL >>>> >>>> >>>> >>>> On 7/8/17, Victor Miller <victorsmiller(a)gmail.com> wrote: >>>>> Yes, I uploaded it. I don't remember where I found it. >>>>> >>>>> Victor >>>>> >>>>> >>>>> On Sat, Jul 8, 2017 at 00:50 Bill Gosper <billgosper(a)gmail.com> wrote: >>>>> >>>>>> https://www.youtube.com/watch?v=9Plt04KpoLc >>>>>> (Could the uploader be our own Victor Miller?) >>>>>> Unfortunately, the video quality is low. The high resolution master >> was >>>>>> discarded by a fly-by-night cassette duplication outfit in Los Angeles. >>>>>> If you find this video boring (as do most?), I think it's because you >>>>>> think >>>>>> you're understanding it, but aren't really. If you're young, come back >>>>>> to >>>>>> it >>>>>> in a couple of years. If you really understand it, it's very >>>>>> interesting. >>>>>> --rwg

2 1

[math-fun] Square root curiosity
by David Wilson 09 Jul '17

09 Jul '17

Let r(a) = sqrt(a). Out of curiosity, I wrote a program to compute, for integer c >= 0 f(c) = (a, b) where integers a, b with 0 <= a <= b minimize r(a) + r(b) > r(c). For example, f(20) = (3, 8) because r(3) + r(8) is the smallest value of the form r(a) + r(b) > r(20). f is not quite well defined because f(24) = (1, 16) = (4, 9) so I proclaim that in ambiguous cases (this is the only one I encountered), smallest a wins, and f(24) = (1, 16). Anyway, I found something interesting. It turns out that f(10^2) = (13, 41) f(10^4) = (1337, 4024) f(10^6) = (236530, 263843) f(10^8) = (10806362, 45060299) There's nothing remarkable about these numbers until we look at their square roots: r(13) = 3.60555+ r(41) = 6.40312+ r(1337) = 36.56501059+ r(4024) = 63.43500610+ r(236530) = 486.34350000796+ r(263843) = 513.65650000754+ r(10806362) = 3287.30315000001178+ r(45060299) = 6712.69685000000577+ Whereas we might reasonably expect the digits to be random, we see curious runs of zeroes. This seems to be related to c being an even power of 10, but I have no explanation. I haven't tried other bases.

1 0

[math-fun] I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial
by Bill Gosper 08 Jul '17

08 Jul '17

Oops, I must have been unclear. Baseball home run trajectories are usually longest with well *below* 45º elevation and lots of backspin. But I was thinking of cannonballs. In a vacuum, 45º is best. With air resistance, best is somewhat lower. But really long range artillery goes farther *above* 45º by getting a few miles up into thinner air. (As North Korea is threatening to demonstrate.) I was speculating that there might be a case where 45º is a local pessimum. --rwg On 2017-07-08 07:15, Fred Lunnon wrote: > This looks a good place to start answering my own question --- > > https://physics.stackexchange.com/questions/200097/flat-throw-vs-45-degree-… > > WFL > > > > On 7/8/17, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: >> Which reminds me of a recent RWG claim that a baseball travels further >> along a horizontal trajectory under gravity at some elevation inceeding >> pi/4 . >> >> How does this come about exactly --- is upward pitching spin involved? >> WFL >> >> >> >> On 7/8/17, Victor Miller <victorsmiller(a)gmail.com> wrote: >>> Yes, I uploaded it. I don't remember where I found it. >>> >>> Victor >>> >>> >>> On Sat, Jul 8, 2017 at 00:50 Bill Gosper <billgosper(a)gmail.com> wrote: >>> >>>> https://www.youtube.com/watch?v=9Plt04KpoLc >>>> (Could the uploader be our own Victor Miller?) >>>> Unfortunately, the video quality is low. The high resolution master was >>>> discarded by a fly-by-night cassette duplication outfit in Los Angeles. >>>> If you find this video boring (as do most?), I think it's because you >>>> think >>>> you're understanding it, but aren't really. If you're young, come back >>>> to >>>> it >>>> in a couple of years. If you really understand it, it's very >>>> interesting. >>>> --rwg

2 1

[math-fun] I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial
by Bill Gosper 08 Jul '17

08 Jul '17

https://www.youtube.com/watch?v=9Plt04KpoLc (Could the uploader be our own Victor Miller?) Unfortunately, the video quality is low. The high resolution master was discarded by a fly-by-night cassette duplication outfit in Los Angeles. If you find this video boring (as do most?), I think it's because you think you're understanding it, but aren't really. If you're young, come back to it in a couple of years. If you really understand it, it's very interesting. --rwg

3 3

[math-fun] A lens of sequential squares
by Ed Pegg Jr 07 Jul '17

07 Jul '17

Take a triangular number and multiply by 4. For example, 45x4=180. Create four copies of all squares up to that number. 4 copies of squares 1-180 gives 720 squares. It is possible to arrange the squares into a lune. http://community.wolfram.com/groups/-/m/t/1141202 has a picture. (△×4)^2 ×4 = lens. Might be called a vesica piscis. --Ed Pegg Jr

1 0

[math-fun] near misses
by Cris Moore 06 Jul '17

06 Jul '17

This is a nice article on "near misses" in mathematics: http://nautil.us/issue/49/the-absurd/the-impossible-mathematics-of-the-real… - Cris

1 0

[math-fun] Mrs. Perkins's Quilt updates
by Ed Pegg Jr 06 Jul '17

06 Jul '17

I have updated http://demonstrations.wolfram.com/MrsPerkinssQuilts/ to 40000, with many new records. The following will need updates. http://www.squaring.net/quilts/mrs-perkins-quilts.html https://oeis.org/A089046 Least edge-length ... Mrs. Perkins's quilt problem. https://oeis.org/A089047 Greatest edge-length ... Mrs. Perkins's quilt problem. ------------- Current A089046: .... 1117, 1473, 1967, 2595, 3465, 4534, 5374, 5498, 5591, 6079, 8794, 14151, 18883, ... Correct A089046: .... 1117, 1485, 1967, 2595, 3465, 4534, 5995, 7907, 10293, 13505, 17785, 23239, 31035, 39571 -------------- Current A089047: .... 4988, 5976, 7640, 9945, 13102, 17304, 23251, 31516, ... Correct A089047: .... 4988, 6443, 8568, 11357, 14877, 19594 ,26697, 34632 --Ed Pegg Jr

3 3

[math-fun] Brocard-like questions
by Dan Asimov 06 Jul '17

06 Jul '17

Brocard's problem (Brocard 1876, 1885; Ramanujan 1913) is to decide whether 4! + 1 = 5^2 5! + 1 = 11^1 7! + 1 = 71^2 are the only factorial-plus-ones equal to a square: (*) n! + 1 = K^2 . It remains unsolved. Question: --------- What is known about generalizing to any exact power: n! + 1 = K^r (r >= 2) and what if we let +1 be replaced by -1 here or in (*) ??? Note: It was shown by Luca (2002) that the abc conjecture (https://en.wikipedia.org/wiki/Abc_conjecture) implies that at most finitely many solutions exist to the still more general equation n! = P(n) where P is any integer polynomial of degree at least 2. —Dan

3 2

[math-fun] I was dropped from the math-fun list from about September 2016 until yesterday
by Neil Sloane 04 Jul '17

04 Jul '17

I was wondering why I hadn't seen any math-fun messages recently, and it turns out that because of a problem with Gmail, or something, I had been dropped from the list nine months ago. I will try to look at the math-fun archives some time, but if there was anything being discussed that had to do with sequences or the OEIS in the past nine months, and you happen to have a copy, please email it to me directly! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane(a)gmail.com

4 3