December 2005 - mathfun@mailman.xmission.com

: [math-fun] Squares on a Cube
by hartmut.holzwart＠allianz.de 26 Dec '05

26 Dec '05

Ich werde ab 24.12.2005 nicht im Büro sein. Ich kehre zurück am 30.12.2005. Ich werde Ihre Nachricht nach meiner Rückkehr beantworten. Ursprüngliches Thema: [math-fun] Squares on a Cube

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Re: [math-fun] Squares on a Cube
by Daniel Asimov 26 Dec '05

26 Dec '05

Christian wrote: << Frank Rubin asks this interesting problem: --------------------------------- "It is possible to write a square on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a square. For example, write 1 on two opposite faces, and 4 on the other 4 faces. The sum of the 3 faces surrounding each vertex is then 1+4+4=9, a square. Can this be done using the squares of 6 distinct whole numbers? If so, what set of 6 such squares has the smallest sum?" . . . >> On the principle that it's always easier to ask more questions than to answer one, how about the other 4 regular polyhedra? How about the 13 Archimedean solids? How about the regular and Archimedean tilings of the plane? And last (and least), how about regular polygons? (I.e., for a given n >= 3, do there exist distinct nonnegative integers {X_k | 0 <= k < n} such that for each index k we have (X_k)^2 + (X_(k+1))^2 is a square number?) -------------------------------------------------------------------------------- And there's no reason not to extend these questions to other regular tilings of surfaces or even higher-dimensional things: E.g., given the tiling of a torus by a rosette of 7 hexagons, same question for the 14 vertices. Or given a 4D regular solid -- say the 24-cell (composed of 24 octahedra, 6 per vertex), can we assign a square number to each of the 24 octahedra such that the sum of the 6 squares in the octahedra containing any vertex v add up to a square number? (Or why stick with only square numbers; why not higher powers?) --Dan

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[math-fun] Squares on a Cube
by Christian Boyer 26 Dec '05

26 Dec '05

Frank Rubin asks this interesting problem: --------------------------------- "It is possible to write a square on each of the 6 faces of a cube (such as a die) so that the 3 faces surrounding each of the 8 vertices sum to a square. For example, write 1 on two opposite faces, and 4 on the other 4 faces. The sum of the 3 faces surrounding each vertex is then 1+4+4=9, a square. Can this be done using the squares of 6 distinct whole numbers? If so, what set of 6 such squares has the smallest sum?" --------------------------------- As for the "3x3 magic squares of squares" unsolved problem, it is a system of 8 equations summing 3 squares. If the die is composed by the 6 integers: a² b², c², d², e² f² (couples a²/f², b²/d² and c²/e² being on opposite faces) the 8 equations are: a²+b²+c² = s² a²+c²+d² = t² a²+d²+e² = u² a²+e²+b² = v² f²+b²+c² = w² f²+c²+d² = x² f²+d²+e² = y² f²+e²+b² = z² I am able to find numerous solutions of 7 equations on 8, for example: 0²+132²+176² = 220² 0²+176²+468² = 500² 0²+468²+351² = 585² 0²+351²+132² = 375² 1200²+132²+176² = 1220² 1200²+176²+468² = 1300² 1200²+468²+351² = 1335² 1200²+351²+132² = 1580625 = 75² x 281 = (1257.229...)² But I am unable to find a complete solution of the 8 equations. Do you see any mathematical impossibility? Christian.

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[math-fun] Casey's theorem
by Fred lunnon 22 Dec '05

22 Dec '05

Around July 1997 under the subject line "geometry puzzle", there was a discussion in math-fun about Ptolemy's and Purser's theorems, which I did my best to derail by whacking Ptolemy into n-space using determinants. I recently came across this thread again, recalled failing to discover what Purser's theorem said, and while repairing the omission encountered Casey's theorem: [See http://mathworld.wolfram.com/CaseysTheorem.html; PursersTheorem.html] Four (oriented) circles in the Euclidean plane are tangent to a fifth (circle or line) just when t_12 t_34 +/- t_13 t_24 +/- t_14 t_23 = 0, where t_ij denotes the length of tangent common to i-th and j-th circles. For this to be interpreted correctly circles and lines must be oriented: think of them as being hairy on one side or the other, the hairyness matching at points of tangency. [Synthetic proofs not being my forte, I leave them to those better equipped.] Casey's theorem follows in a way similar to Ptolemy's, from the case n = 2 of another beautiful determinant which I didn't mention previously: Given n+4 oriented spheres in Euclidean n-space, the determinant of their common tangent squared lengths has value zero: | (t_ij)^2 | = 0. There is a puzzling feature of Casey's theorem which seems to have gone unremarked. There are two distinct ways in which a fifth (oriented) circle may be tangent to four others: e.g. in the simplest case where all four are external to each other and oriented alike, a tangent circle may be either surrounded by them or surrounding them; but the Casey criterion fails to distinguish these alternatives. How may the two cases be distinguished? More interestingly, what is an analogous criterion for four circles to be tangent to two others? Fred Lunnon 23/12/05

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[math-fun] Normality of sqrt(N)
by Schroeppel, Richard 22 Dec '05

22 Dec '05

A shocking abstract: http://arxiv.org/abs/math/0512404 On the simple normality to base 2 of the square root of s, for s not a perfect square. Richard Isaac Lehman College and Graduate Center, CUNY 13 pages We show that each number of the form, the square root of s for s not a perfect square, is simply normal to the base 2. The argument uses some elementary ideas from the calculus of finite differences. ---- My comments [Rich]: "Simply Normal" means only individual bit statistics are considered, not groups of bits. This comes down to 0 and 1 each being 50% in the limit. The author says at the end of the paper he thinks he can get "Normal". The method seems to be restricted to base 2, and depends on "there are only two different digits to consider". It also depends on the algebraic degree being 2 for the number being examined, although there might be some wiggle room here, with pseudo- Fibonacci-ratio numbers like x in x^3 = x+1 being accessible. The only previous result I'm aware of is trivial remark that the first N bits of sqrt2 must have at least O(sqrtN) 1 bits, simply to get enough cross product terms in the square to make the 1s in 1.11111111... . I looked over the paper, but can't figure out the hard part. It seems to have some good ideas, but I'm skeptical. For S=2, i.e. normality of sqrt2, his argument looks at the relationship between u = sqrt2 - 1 = .4142135... and v = u^2 = 3 - 2 sqrt2 = 1 - 2u = .1715728... Since v = 1-2u, the bit patterns for u and v are complementary and shifted one place. He somehow relates this to the relationship between (the bit pattern of) a number and its square. He looks at the temporary, diminishing, effect of a bit position in v, where a bit has only the effect 1/N in the average bit value of the first N bits of v, while the complementary shifted bit in u has a more permanent effect on the average bit value of u^2. The overall argument puzzles me, since I would expect to see more complicated machinery that looked at carries between bit positions, and at dot products of bit patterns. The exception for s a perfect square enters into the proof as simply that non-square s have non-terminating binary expansions. Rich

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[math-fun] Codes - note from Berlekamp
by Schroeppel, Richard 21 Dec '05

21 Dec '05

I received the note below from Berlekamp, about recent developments in Error Correcting Codes. Posted with his permission. --Rich -------------- (from Elwyn Berlekamp, Nov 22 2005) Coding theory remains an active and dynamic subject, as it has been for the past 58 years. Although I haven't published any papers in this area since 1996, I remain an interested spectator. Low Density Parity Check Codes were invented and studied by Bob Gallager in his PhD thesis in the early 1960s. Even now, most of what is known about them was published by him then. He became an assistant professor at MIT soon after he finished his PhD. Coding theory was then very active at MIT. I was Gallager's second PhD student. Shannon, Elias, and Wozencraft were also on my committee. No one favored further work on low density parity check codes at that time, because they were felt to be relatively well understood but impractical. Convolutional codes with sequential decoding enjoyed popularity in the early and mid 1960s. In the 1970s, Viterbi decoding became more popular, especially in military voice communication systems. Reed-Solomon codes, mostly using decoding algorithms with which I was personally involved, also became popular, especially in computer memory devices such as magnetic discs, optical discs, and magnetic tapes. They are now enjoy very widespread use. Turbo codes appeared in the late 1990s. A revival of interest in low density parity check codes began about the same time. It is true that, for certain channels including the white Gaussian noise channel, both turbo codes and low density parity check codes can achieve performance much closer to the Shannon capacity than any of their competitors. Each is now the topic of many papers. Many theorists study turbo codes in an effort to better understand their good performance, which was initially demonstrated almost entirely empirically. There are some system environments in which one or the other of these two types of coding systems is the clear winner. It is also true that most communication and computer memory systems, even new ones, continue to use older coding techniques such as Reed-Solomon. That's because real systems require tradeoffs among many parameters. In 1980 I published an overview entitled "The Technology of Error-Correcting Codes" in the Proceedings of the IEEE. Most of the considerations discussed there, including code rate, block length, latency, and interface issues, remain highly relevant. The advantages of low density parity check codes become decisive only at VERY long block lengths. A decade or two ago, memory was viewed as too expensive. Latency has now become a bigger concern. Turbo codes require considerably more decoding computation than Viterbi codes or RS codes. The relative performance advantage/disadvantage is strongly dependant on the code rate, and for various reasons not directly related to the coding, most memory systems use very high-rate codes, an environment which is especially favorable to Reed-Solomon. So I would say that the performance claims of turbo codes and low density parity check codes are true, but their impact is often exaggerated. Another major development of the 1990s is the work of Madhu Sudhan and others on list decoding of low-rate Reed-Solomon codes. This buys improved RS performance at the cost of more computation. It's primary use, to date, has been in computer science programs doing probabilistic search rather than in any of the more traditional applications of coding theory.

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[math-fun] minimising leaf-count
by wouter meeussen 21 Dec '05

21 Dec '05

It is possible to write any integer as sums of expressions containing only digits 1 or 2 or powers of 2, so that the exponents are also such expressions. Cfr Bill Dubuque's funny transliterations a few years back. But is it feasible to minimise the 'leaf count' of the result by expressing {n} as {2^k}-{n-2^k} if the count of 1-bits exceeds the count of 0-bits in the binary representation of n? So, 14 becomes -2+2^(2^2) and not just 2+2^2+2^(1+2) What I aim for is an over-all minimised leaf-count, not a minimisation at each step. a trial on n= 30197521 could give: 1 + 2^2^2 + 2^2^(1 + 2) + 2^(-1 + 2^2^2) + 2^(2 + 2^2^2) + 2^(1 + 2 + 2^2^2) + 2^(2 + 2^2 + 2^2^2) + 2^(1 + 2^(1 + 2)) + 2^(2 + 2^(1 + 2)) + 2^(2 + 2^2 + 2^(1 + 2)) + 2^(2^2^2 + 2^(1 + 2)) + 2^(-1 - 2^(1 + 2) + 2^(1 + 2^2)) but I doubt if it is optimal. W.

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[math-fun] yet an other Mersenne prime?
by wouter meeussen 18 Dec '05

18 Dec '05

keep an eye on http://mersenne.org/primenet/ a (as yet unconfirmed) candidate M43 between 25,000,000 and 33,000,000 (or abouts) seems to be in the verification stage. (found on http://www.mersenneforum.org/) ------- Mersenne Exponent Test State ------- Assigned in Tests Cleared Since Last Synchronization Factoring only : 9620 Factored composite : 860 Lucas-Lehmer testing : 56487 Lucas-Lehmer composite: 4577 Double-checking LL : 20150 Double-checked LL : 2671 Prime, UNVERIFIED : 1 ---------------------- ------- ---------------------- ------- TOTAL : 86257 TOTAL : 8109 'recent' finds: February 18, 2005: Prime M(25964951) May 15, 2004: Prime M(24036583) November 17, 2003: Prime M(20996011) W.

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[math-fun] What's the cross product of two mathematicians?
by Schroeppel, Richard 17 Dec '05

17 Dec '05

Congratulations to Scott & Cheryl Beaver on their early Christmas present! --Rich ----- Cheryl writes We are happy to announce the arrival of baby Evan Walter Beaver on Wednesday December 14, 2005 at 8:51 am. Evan weighed in at 7 lbs. 5 oz and 20 1/4 inches long. Everything went great and the whole family is doing fine.

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[math-fun] 2F1(golden ratio)
by R. William Gosper 16 Dec '05

16 Dec '05

I have the weird feeling I once knew this and forgot it, but since I no longer have a decent message massager (or "masausage", as a young friend recently misspoke) we'll have to risk repetition. 1 - sqrt(5) hyper_2f1(a, 5 a + 1, 3 a + 1, -----------) = 2 5 a --- 2 3 2 pi 5 (a - -)! (a - -)! 4 %pi cos(--) 5 5 10 (3 a)! (---------------------- + ----------------------------) pi 5 a (5 a)! 4 %pi cos(--) --- 10 2 3 2 5 a! 5 (a - -)! (a - -)! 5 5 Note G/(5a)! + 1/G/5a!. I think this is almost, but not quite derivable from the formulas in A&S Chapter 15, starting with 15.1.31, and then various linear and quadratic transformations. The obstacle seems to be A&S's negligence to provide contiguous pairs both for 15.1.31 and the quadratic transformations. The latter can be somewhat tediously derived from the three-term recurrences, but 15.1.31 seems to be cursed with some sort of singularity that thwarts population of the contiguity grid starting with F(a) and F(a+-3). (Of course, the good way to do things is with 3x3 matrices that have the contiguous pair built in. I'm belatedly nagging the DLMF people about this.) There are also similar identities for 2F1(phi^-2). Once all this goes into the HYPERSIMP facility, we'll see whether 2F1(a,5a,3a,1/phi) or some contiguous neighbor has a monomial rhs. --rwg

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