Playing with numbers : puzzles for the daily maths lesson.

Ofrece actividades fotocopiables, juegos y rompecabezas para los alumnos de la etapa clave 2 (key stage 2) del currículo nacional de Inglaterra y Gales, es decir para el nivel de primaria. Están diseñadas para desarrollar en los niños la comprensión y la facilidad con los números y, así, proporcionarles una buena base para el desarrollo de sus habilidades matemáticas. Las actividades se clasifican en tres secciones para los grupos de edad de 7 a 9, de 8 a 10 y de 9 a 11.

Rational modification of the hierarchy of intermolecular interactions in molecular crystal structures by using tunable halogen bonds

A family of 16 isomolecular salts (3-XpyH)(2)[MX'(4)] (3-XpyH=3-halopyridinium; M=Co, Zn; X=(F), Cl, Br, (I); X'=Cl, Br, I) each containing rigid organic cations and tetrahedral halometallate anions has been prepared and characterized by X-ray single crystal and/or powder diffraction. Their crystal structures reflect the competition and cooperation between non-covalent interactions: N-H center dot center dot center dot X'-M hydrogen bonds, C-X center dot center dot center dot X'-M halogen bonds and pi-pi stacking. The latter are essentially unchanged in strength across the series, but both halogen bonds and hydrogen bonds are modified in strength upon changing the halogens involved. Changing the organic halogen (X) from F to I strengthens the C-X center dot center dot center dot X'-M halogen bonds, whereas an analogous change of the inorganic halogen (X') weakens both halogen bonds and N-H center dot center dot center dot X'-M hydrogen bonds. By so tuning the strength of the putative halogen bonds from repulsive to weak to moderately strong attractive interactions, the hierarchy of the interactions has been modified rationally leading to systematic changes in crystal packing. Three classes of crystal structure are obtained. In type A (C-F center dot center dot center dot X'-M) halogen bonds are absent. The structure is directed by N-H center dot center dot center dot X'-M hydrogen bonds and pi-stacking interactions. In type B structures, involving small organic halogens (X) and large inorganic halogens (X'), long (weak) C-X center dot center dot center dot X'-M interactions are observed with type I halogen-halogen interaction geometries (C-X center dot center dot center dot X' approximate to X center dot center dot center dot X'-M approximate to 155 degrees), but hydrogen bonds still dominate. Thus, minor but quite significant perturbations from the type A structure arise. In type C, involving larger organic halogens (X) and smaller inorganic halogens (X'), stronger halogen bonds are formed with a type II halogen-halogen interaction geometry (C-X center dot center dot center dot X' approximate to 180 degrees; X center dot center dot center dot X'-M approximate to 110 degrees) that is electrostatically attractive. The halogen bonds play a major role alongside hydrogen bonds in directing the type C structures, which as a result are quite different from type A and B.

Primality-testing Mersenne numbers

Report of a systematic review of Mersenne numbers 2^p-1 for p < 62982.

Mersenne numbers

These notes have been issued on a small scale in 1983 and 1987 and on request at other times. This issue follows two items of news. First, WaIter Colquitt and Luther Welsh found the 'missed' Mersenne prime M110503 and advanced the frontier of complete Mp-testing to 139,267. In so doing, they terminated Slowinski's significant string of four consecutive Mersenne primes. Secondly, a team of five established a non-Mersenne number as the largest known prime. This result terminated the 1952-89 reign of Mersenne primes. All the original Mersenne numbers with p < 258 were factorised some time ago. The Sandia Laboratories team of Davis, Holdridge & Simmons with some little assistance from a CRAY machine cracked M211 in 1983 and M251 in 1984. They contributed their results to the 'Cunningham Project', care of Sam Wagstaff. That project is now moving apace thanks to developments in technology, factorisation and primality testing. New levels of computer power and new computer architectures motivated by the open-ended promise of parallelism are now available. Once again, the suppliers may be offering free buildings with the computer. However, the Sandia '84 CRAY-l implementation of the quadratic-sieve method is now outpowered by the number-field sieve technique. This is deployed on either purpose-built hardware or large syndicates, even distributed world-wide, of collaborating standard processors. New factorisation techniques of both special and general applicability have been defined and deployed. The elliptic-curve method finds large factors with helpful properties while the number-field sieve approach is breaking down composites with over one hundred digits. The material is updated on an occasional basis to follow the latest developments in primality-testing large Mp and factorising smaller Mp; all dates derive from the published literature or referenced private communications. Minor corrections, additions and changes merely advance the issue number after the decimal point. The reader is invited to report any errors and omissions that have escaped the proof-reading, to answer the unresolved questions noted and to suggest additional material associated with this subject.

Primality-testing Mersenne Numbers (II)

Reports the factor-filtering and primality-testing of Mersenne Numbers Mp for p < 100000, the latter using the ICL 'DAP' Distributed Array Processor.

Mersenne Numbers: consolidated results

This document provides and comments on the results of the Lucas-Lehmer testing and/or partial factorisation of all Mersenne Numbers Mp = 2^p-1 where p is prime and less than 100,000. Previous computations have either been confirmed or corrected. The LLT computations on the ICL DAP is the first implementation of Fast-Fermat-Number-Transform multiplication in connection with Mersenne Number testing. This paper championed the disciplines of systematically testing the Mp, and of double-sourcing results which were not manifestly correct. Both disciplines were adopted by the later GIMPS initiative, the 'Great Internet Mersenne Prime Search, which was itself one of the first web-based distributed-community projects.

Rational conversion of substrate and product specificity in a Salvia monoterpene synthase: structural insights into the evolution of terpene synthase function

Terpene synthases are responsible for the biosynthesis of the complex chemical defense arsenal of plants and microorganisms. How do these enzymes, which all appear to share a common terpene synthase fold, specify the many different products made almost entirely from one of only three substrates? Elucidation of the structure of 1,8-cineole synthase from Salvia fruticosa (Sf-CinS1) combined with analysis of functional and phylogenetic relationships of enzymes within Salvia species identified active-site residues responsible for product specificity. Thus, Sf-CinS1 was successfully converted to a sabinene synthase with a minimum number of rationally predicted substitutions, while identification of the Asn side chain essential for water activation introduced 1,8-cineole and alpha-terpineol activity to Salvia pomifera sabinene synthase. A major contribution to product specificity in Sf-CinS1 appears to come from a local deformation within one of the helices forming the active site. This deformation is observed in all other mono- or sesquiterpene structures available, pointing to a conserved mechanism. Moreover, a single amino acid substitution enlarged the active-site cavity enough to accommodate the larger farnesyl pyrophosphate substrate and led to the efficient synthesis of sesquiterpenes, while alternate single substitutions of this critical amino acid yielded five additional terpene synthases.

The relationships of time and temperature to body weight and numbers of endospores in Pasteuria penetrans-infected Meloidogyne javanica females

Tomato plants inoculated with Meloidogyne javanica juveniles infected with Pasteuria penetrans were grown in a glasshouse (20-32degreesC) for 36, 53, 71 and 88 days and in a growth room (26-29degreesC) for 36, 53, 71 and 80 days. Over these periods the numbers of P penetrans endospores in infected M. javanica females and the weights of individual infected females increased. In the growth room, most spores (2.03 x 10(6)) were found after 71 days. However, in the glasshouse the rate of increase was slower and spore numbers were still increasing at the final sampling at 88 days (2.04 x 10(6)), as was the weight of the nematodes (72 mug). Weights of uninfected females reached a maximum of 36.2 and 43.1 mug after 71 days in the growth room and glasshouse, respectively.

Estimating numbers of infectious units from serial dilution assays

The paper concerns the design and analysis of serial dilution assays to estimate the infectivity of a sample of tissue when it is assumed that the sample contains a finite number of indivisible infectious units such that a subsample will be infectious if it contains one or more of these units. The aim of the study is to estimate the number of infectious units in the original sample. The standard approach to the analysis of data from such a study is based on the assumption of independence of aliquots both at the same dilution level and at different dilution levels, so that the numbers of infectious units in the aliquots follow independent Poisson distributions. An alternative approach is based on calculation of the expected value of the total number of samples tested that are not infectious. We derive the likelihood for the data on the basis of the discrete number of infectious units, enabling calculation of the maximum likelihood estimate and likelihood-based confidence intervals. We use the exact probabilities that are obtained to compare the maximum likelihood estimate with those given by the other methods in terms of bias and standard error and to compare the coverage of the confidence intervals. We show that the methods have very similar properties and conclude that for practical use the method that is based on the Poisson assumption is to be recommended, since it can be implemented by using standard statistical software. Finally we consider the design of serial dilution assays, concluding that it is important that neither the dilution factor nor the number of samples that remain untested should be too large.