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SummaryLatin squares form the basis for the recreational puzzles sudoku and KenKen. In this article we show how useful several ideas from number theory are in solving a KenKen puzzle. For example, the simple notion of triangular number is surprisingly effective. We also introduce a variation of KenKen that uses the Gaussian integers in order to illustrate the concept of unique factorization.

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The kinetics of ion pairing of IrCl 6 3 - with K+, Ca2+ and tetraethylammonium (TEA+) were investigated by steady-state voltammetry at disk and hemispherically shaped Pt electrodes. A general square-scheme mechanism is employed that allows for redox reactions of both free and ion paired IrCl 6 3 - and IrCl 6 2 - . Numerical simulations of the voltammetric response employing the ion pairing square scheme suggest that the dissociation rate constant of the reactant ion pair can be determined from the voltammetric wave shape recorded at very small electrodes. No ion-pair kinetic limitation was discernable in K+ and Ca2+ solutions using Pt hemispherical electrodes with radii as small as 13 nm, indicating that ion-pair dissociation in these solutions occurs at rates greater than 107 s−1. However, the voltammetric wave in 1 M TEA+ displays a shoulder at electrodes with radii less than 500 nm, consistent with a slower rate of ion-pair dissociation. The appearance of the shoulder is predicted by the numerical simulations and corresponds to a direct observation of the oxidation of both free and ion paired IrCl 6 3 - in the same solution.

The steady-state voltammetric oxidation of hexachloroiridate(III), IrCl6(3-) (1-5 mM), in the presence and absence of an excess supporting electrolyte was investigated at disk- and hemispherical-shaped Pt electrodes with radii ranging from 48 nm to 12.5 microm. Thermodynamic, kinetic, and transport parameters that define the shape and magnitude of the voltammetric wave exhibit a complex dependence on whether a supporting electrolyte is present in the solution. First, the half-wave potential, E1/2, for oxidation of IrCl6(3-) shifts to more positive potentials in the presence of a supporting electrolyte, a consequence of the relative difference in the strength of ion pairing of IrCl6(3-) and IrCl6(2-) by the supporting electrolyte cation. E1/2 increases in the order no electrolyte < n-tetrabutylammonium < Na+ approximately K+ approximately Ca2+, but is independent of the supporting electrolyte anion (Cl-, NO3-, PF6-). Second, the heterogeneous electron-transfer rate constant for oxidation of IrCl6(3-) increases by approximately an order of magnitude in the presence of a supporting electrolyte. Third, in the absence of electrolyte, mass transport limited currents deviate significantly from predicted values based on the Nernst-Planck equation, but only when the electrode radius is smaller than ca. 1 microm. The latter two effects (Frumkin and dynamic diffuse layer effects) result from the dependence of interfacial electrical fields and, thus, the rates of electron-transfer and ion migration, on the supporting electrolyte concentration. We also demonstrate that the theoretical shape of the voltammetric response for oxidation or reduction of a highly charged redox species (e.g., IrCl6(3-)) is essentially independent of whether a supporting electrolyte is present in the solution. This finding can greatly simplify the analysis of heterogeneous electron-transfer rates using steady-state voltammetry in low ionic strength solutions.

The characterization of quasi-hemispherical Pt electrodes of nanometer dimensions (radius 2-150 nm), prepared by electrophoretic coating of etched Pt wires with poly-(acrylic acid), is described. The goals of these experiments are to estimate the accuracy of using steady-state voltammetric limiting currents (i(lim)) in determining the true electrode area and to develop new electrochemical methods for rapidly screening individual electrodes for non-ideal geometries. Electrochemical active areas were determined by measuring the electrical charge (Q) associated with oxidation of adsorbed bis(2,2'-bipyridine)chloro(4,4'-trimethylenedipyridine)osmium(II) in fast-scan voltammetric measurements (scan rate 1000 V/s). Voltammetric peaks corresponding to oxidation of as few as approximately 7000 molecules (approximately 11 zmol) at individual electrodes are reported, allowing precise measurement of electrode areas as small as approximately 10(-10) cm2. A plot of i(lim) (for a soluble redox species) versus Q1/2 (for an adsorbed redox species), constructed from i(lim)-Q1/2 data pairs obtained as a function of the electrode radius, is shown to be linear if the electrode geometry is independent of electrode radius; departure of experimental values from the straight-line plot is a diagnostic indicator of a nonideal electrode geometry. The results indicate that approximately 50% of the electrodes prepared by the electrophoretic polymer-coating procedure are quasi-hemispherical, the remaining being recessed slightly below the polymer coating. The heterogeneous electron-transfer rate constant for the oxidation of the ferrocenylmethyltrimethylammonium cation in H2O/ 0.2 M KCl was also determined from steady-state voltammetry using the method of Mirkin and Bard and found to be 4.(8) +/- 3.(2) cm/s with alpha = 0.6(4) +/- 0.1(5).

The knight is the only piece in chess that does not move in a straight line. Instead, it moves in an "L"-two squares in either a vertical or horizontal direction and then one square in a perpendicular direction. It is the strangeness of this move that has made the Knight's Tour Problemn one of the most intriguing in all of recreational mathematics: Can a knight visit each square of a chessboard by a sequence of knight's moves, andfinish on the same square as it began? Since a chessboard can be represented as a graph in which each vertex corresponds to a square, and edges correspond to those pairs of squares connected by a knight's move (FIGURE 1 illustrates this for a 4 x 4 board), finding a knight's tour amounts to finding a Hamiltonian cycle in the corresponding graph, a notoriously difficult general problem in graph theory (see [5]). However, we can easily see that there is no knight's tour for a 4 x 4 board since any Hamiltonian cycle would have to include the four edges incident to the two corner vertices indicated in FIGURE 1; this is impossible since these four edges already form a cycle that includes only four vertices.

INTRODUCTION Two Thousand Years of Combinatorics PART I: ANCIENT COMBINATORICS 1. Indian Combinatorics 2. China 3. Islamic Combinatorics 4. Jewish Combinatorics 5. Renaissance Combinatorics 6. The Origins of Modern Combinatorics 7. The Arithmetical Triangle PART II: MODERN COMBINATORICS 8. Early Graph Theory 9. Partitions 10. Block Designs 11. Latin Squares 12. Enumeration (18th-20th Centuries) 13. Combinatorial Set Theory 14. Modern Graph Theory AFTERMATH A Personal View of Combinatroics

Purpose: Under normal conditions genomic CpG islands are not methylated. Hypermethylation of a CpG island in the 5' regulatory region of a gene has the capacity to silence gene transcription. Recently, hypermethylation of a CpG island at D17S5 on chromosome 17P13.3 has been shown to be a frequent tumor-specific event. When it has been observed, hypermethylation of D17S5 occurs solely in neoplastic tissues. Consequently, it has been hypothesized that hypermethylation of D17S5 may be an important carcinogenic event in the organs in which it occurs (colon, kidney, and brain). In this study we examine D17S5 hypermethylation in DNA from the prostate, a gland which is unique in that it undergoes hyperplastic or neoplastic growth or both in virtually all aging men.Materials and Methods: The methylation sensitive restriction enzyme Notl, a cDNA probe specific for the D17S5 locus, and Southern blotting were used to assay for hypermethylation of D17S5 in DNA derived from normal, benign hyperplastic and mali...

The resistive-pulse method was used to monitor the pressure-driven translocation of multilamellar liposomes with radii between 190 and 450 nm through a single conical nanopore embedded in a glass membrane. Liposomes (0% and 5% 1,2-dioleoyl-sn-glycero-3-phospho-l-serine (sodium salt) in 1,2-dilauroyl-sn-glycero-3-phosphocholine or 0%, 5%, and 9% 1,2-dipalmitoyl-sn-glycero-3-phospho(1'-rac-glycerol) (sodium salt) in 1,2-dipalmitoyl-sn-glycero-3-phosphocholine) were prepared by extrusion through a polycarbonate membrane. Liposome translocation through a glass nanopore was studied as a function of nanopore size and the temperature relative to the lipid bilayer transition temperature, T(c). All translocation events through pores larger than the liposome, regardless of temperature, show translocation times between 30 and 300 μs and current pulse heights between 0.2% and 15% from the open pore baseline. However, liposomes at temperatures below the T(c) were captured at the pore orifice when translocation was attempted through pores of smaller dimensions, but squeezed through the same pores when the temperature was raised above T(c). The results provide insights into the deformation and translocation of individual liposomes through a porous material.