October 2003 - mathfun@mailman.xmission.com

[math-fun] Shortest successful doctoral thesis in Mathematics?
by Guy Haworth 24 Oct '03

24 Oct '03

A question arose the other day as to what is the shortest successful doctoral thesis in Mathematics. I start the bidding with Walter Boas' 15pp thesis of 1930. Guy Haworth

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[math-fun] a dozen problems involving packing integer squares in squares
by Erich Friedman 23 Oct '03

23 Oct '03

greetings. since there has some been some discussion recently about the mrs. perkins quilt problem, i thought i would share some other open (as far as i know) problems involving tiling integer squares in squares. if you can add to my knowledge on any of these, it would be greatly appreciated. ------------------------------------------------------------------- 1) mrs. perkins quilt problem: pack a square of side n with the fewest number f(n) of co-prime squares. recently lance gay has shown f(n) <= 19 for 101 <= n <= 108. it is fairly easy to show that f(n) <= 20 for 109 <= n <= 132. this is problem C3 in UPIG. ------------------------------------------------------------------- 2) pack a square of side n with squares so that the smallest square is as large as possible. some partial results are at http://www.stetson.edu/~efriedma/mathmagic/1298.html. the values listed there are known to be best, by computer search. can anyone extend these values? it appears that the smallest square grows like n^(1/2). i had a custom wood puzzle made of a square of side 103 cut into smaller squares all of which are 15 and larger. ------------------------------------------------------------------- 3) can every sufficiently large square be cut into squares with no size square being used more than twice? it appears true for squares of size 22 and more, but seems difficult to prove. perhaps it is even true that all sufficiently large squares admit perfect tilings, so that only one square of each size is needed? ------------------------------------------------------------------- 4) partridge packings: can 1 square of side 1, 2 squares of side 2, up to n squares of side n, be packed into a square of side n(n+1)/2 ? it is not possible for 2 <= n <= 7, and it is possible for 8 <= n <= 34. is it possible for all larger n? ------------------------------------------------------------------- 5) what is the smallest value of n for which n squares each of sides 1-24 can pack n squares of side 70? it is well known that n=1 doesn't work. it must be easy to prove that this is possible for large n. what is the smallest value? i managed to pack 4 copies of 1-24 inside a square of side 140. ------------------------------------------------------------------- 6) diverse packings: what squares can be packed with 1 or 2 squares each of sides 1-n, for some n? some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/0602.html. it is conjectured that this is possible for n >= 34, but existence of infinitely many packings of this sort is still in doubt. do packings of this sort exist for every n >= 9? ------------------------------------------------------------------- 7) square strip packings: which squares can be packed with "strips" of squares of width 1-n, for some n? some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/1201.html. (a strip is a horizontally or vertically adjacent union of squares of a given size, so a 3x12 rectangle is a strip of width 3.) it seems clear that all squares of size 6 or more can be packed in this way, but can anyone prove this? ------------------------------------------------------------------- 8) decending squares: which squares can be packed with squares so that every square directly above a square is strictly smaller? the answer seems to be all except 2,4,5,7,8,9,11,13,15,17,19,21,23,29,33. is this correct? can anyone prove it? some of these make fabulous puzzles. ------------------------------------------------------------------- 9) anti-partridge packings: can n squares of side 1, (n-1) squares of side 2, up to 1 square of side n be packed inside a square without overlap? packings of n=6 and n=25 are given at: http://www.stetson.edu/~efriedma/anti/. this is only possible for certain values of n due to area considerations. the next possible case is n=96. is it possible? ------------------------------------------------------------------- since i didn't quite make a dozen, here are a few packing (rather than tiling) problems. ------------------------------------------------------------------- 10) can you prove the smallest square that can contain 12 non-overlapping unit squares is side 4? it is almost certainly true, and has been proved for 14 unit squares. this is part of problem D4 in UPIG. ------------------------------------------------------------------- 11) given integers n and m, find n squares of one size and m squares of another (possibly equal) size so that these n+m non-overlapping squares fit inside a unit square and have the largest possible area. some partial results are at: http://www.stetson.edu/~efriedma/mathmagic/0400.html. some of the best known results involve tilted squares. surely some of these can be improved. ------------------------------------------------------------------- 12) what is the smallest square f(n) that contains squares of sides 1-n? does anyone even have a list of the best known values for f(n)? this is part of problem D5 in UPIG. ------------------------------------------------------------------- erich friedman

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[math-fun] Free "Readers's Digest" version of experimental math books
by David H Bailey 23 Oct '03

23 Oct '03

As some of you know, Jon Borwein, myself and (for vol 2) Roland Girgensohn have completed a two-volume work on computational/experimental mathematics. Vol 1, "Mathematics by Experiment: Plausible Reasoning on the 21st Century", should be available (eg, from Amazon) in 2-3 weeks. Vol 2, "Experimentation in Mathematics: The Computational Road to Discovery", should be availble in 2 months or so. In the meantime, we are pleased to announce that a 75-page "Reader Digest" condensation of these two books is now available FREE at the website http://www.expmath.info David H Bailey david(a)dhbailey.com P.S. I believe I mentioned this on Math-Fun a month or so ago, but it was during the "SoBig" spam blizzard, so some of you might have missed it.

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Fwd: Re: [math-fun] ANSWER Insomniacs Problem Revisited:
by otto2hoh 23 Oct '03

23 Oct '03

On Monday 20 October 2003 07:55 am, you wrote: > Otto, > I somehow missed the original posing of this problem. The problem > statement makes a point of identifying your physical location, but I > fail to see how that could impact the solution. Could you please tell me > if this is relevant? Yes.. Because daylight savings does not exist everywhere. Local time implies the daylight savings change. That is also why I said approximately one year. The change occurs in the spring, and the repeat does not occur until the next spring. The day for the change is chosen according to the week and so the number of seconds, even without taking leap years and leap seconds into account is variable. As I said, the problem is more tricky; I didn't claim it was better. It occurred to me because it is what I assumed the answer to the original Conway problem was going to be. Actually I was a little disappointed when the original problem was easily solved with a seven line program. Regards Otto otto(a)olympus.net Regards Otto otto(a)olympus.net > Regards, > Nick > > otto wrote: > >>John Conway posted the original insomniacs problem below, which > >>assumes a 12 hour clock with no leading zeros. > >> > >>The following trickier problem also assumes a 12 digit clock. > >>""" > >>Here in washington state I am watching palindromic numbers > >>go by in local time on my on my computers 12-hour clock without > >>leading zeros. I count 281 seconds between two palindromes. How > >>long will it be before I count 281 seconds between two numbers again? > > > >The answer is approximately one year. > > > >Regards > >Otto > > > >John Conway's Original Problem: > >> There's a digital clock at my bedside that shows > >>hours minutes and seconds, and which I usually look > >>at the moment I wake up. A few days ago, I happened > >>to wake up at a palindromic time, and out of interest, > >>waited until the next palindromic time, and noticed the > >>interval between them. > >> > >> Last night exactly the same thing happened, but the > >>interval was exactly four times as long as in the previous > >>case. What did the clock read when I woke up last night? > >> > >> John Conway > > > >------------------------------------------------------- > > > >------------------------------------------------------- > > > >_______________________________________________ > >math-fun mailing list > >math-fun(a)mailman.xmission.com > >http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -------------------------------------------------------

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[math-fun] Axioms and logic
by Jon Perry 22 Oct '03

22 Oct '03

As my good friend Dr. C. once reminded me, The logical existence argument for axioms is preposterous or true... I hit him... Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] paradoxes
by Jon Perry 22 Oct '03

22 Oct '03

Another wow, 'We live for the paradox' Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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Re: [math-fun] greatest integer
by N. J. A. Sloane 22 Oct '03

22 Oct '03

Mike Stay asked: Are there any useful definitions of greatest integer function for algebraic integers? Me: It might be better to look for analogues of the "round to nearest integer" function. Then there's an obvious candidate: tile the space with Voronoi cells centered at the integers, and map any point in the cell to its center. There are fast algorithms too - see Chap 21 of SPLAG (Sphere Packings, Lattices and Groups, by JHC and me). Neil Sloane

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[math-fun] Mr. Perkins Quilt
by Jon Perry 22 Oct '03

22 Oct '03

This MPQ talk is similar to a problem I had considered - what is the smallest square needed to house all 1..n squares? e.g. with 1x1, we need a 1x1 square with 1x1 and 2x2, we need a 3x3 square with 1x1, 2x2 and 3x3, we need a 5x5 square. So a lower bound is 2n-1. However with all the squares from say 1 to 8, this packing fails. Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] plane filling ...
by Joerg Arndt 21 Oct '03

21 Oct '03

Hope it's ok to send a few lines of pari/gp code: /* functions correct to order = 4^N-1 */ fpp(y)= { local( t, Y ); t = 1; Y=y; for (k=1, N, t *= (+1+Y+Y^2-Y^3); Y=Y^4; ); return( t-1 ); } fmm(y)= { local( t, Y ); t = 1; Y=y; for (k=1, N, t*= (-1+Y+Y^2+Y^3); Y=Y^4; ); return( t*y-Y ); } hhpm(y)= { local( tp, tm ); tp = fpp(y); tm = fmm(y); return( ((tp+tm) + I*(tp-tm))/2 ); } start gp, copy & paste above into session ... ? N=4; ? taylor(hhpm(y), y) y + I*y^2 - y^3 + I*y^4 + I*y^5 + y^6 - I*y^7 + y^8 + I*y^9 + y^10 - I*y^11 - I*y^12 - y^13 - I*y^14 + y^15 + O(y^16) Now start at the origin and for each coefficient draw a one-unit line to the right/left/up/down when the coefficient is +1/-1/+I/-I. ... or use: ? taylor(hhpm(y)/(1-y), y) y + (1 + I)*y^2 + I*y^3 + 2*I*y^4 + 3*I*y^5 + (1 + 3*I)*y^6 + (1 + 2*I)*y^7 + (2 + 2*I)*y^8 + (2 + 3*I)*y^9 + (3 + 3*I)*y^10 + (3 + 2*I)*y^11 + (3 + I)*y^12 + (2 + I)*y^13 + 2*y^14 + 3*y^15 + O(y^16) That is connect the points in the complex plane (given as coefficients). The following functions gives the turns of the above curve: hht(y)= { /* correct to order = 4^N-1 */ local( t, Y, F, s, p ); t = 1; Y=y; p = 1; F = y + y^2; for(k=2, N, Y = Y^4; t = -F + Y*F + Y^2*F - Y^3*F; p *= 4; if ( 0==(k%2), t += y^(1*p-1); t += y^(3*p); t -= (y+1)*y^(2*p-1); , /* else */ t += y^(1*p); t += y^(3*p-1); ); F = t; ); return( F-(N%2) ); } If you see how to make the latter algorithm more regular, please let me now. Trivial? Well-known? Nice? Boring? Fun? -- p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 Life is hard and then you die.

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[math-fun] Golden Pearls
by Jon Perry 21 Oct '03

21 Oct '03

Plz see; http://www.users.globalnet.co.uk/~perry/maths/wow/wow.htm for true universal enlightment. Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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