Using the Tower of Hanoi puzzle to infuse your mathematics classroom with computer science concepts

This article suggests that logic puzzles, such as the well-known Tower of Hanoi puzzle, can be used to introduce computer science concepts to mathematics students of all ages. Mathematics teachers introduce their students to computer science concepts that are enacted spontaneously and subconsciously throughout the solution to the Tower of Hanoi puzzle. These concepts include, but are not limited to, conditionals, iteration, and recursion. Lessons, such as the one proposed in this article, are easily implementable in mathematics classrooms and extracurricular programmes as they are good candidates for ‘drop in’ lessons that do not need to fit into any particular place in the typical curriculum sequence. As an example for readers, the author describes how she used the puzzle in her own Number Sense and Logic course during the federally funded Upward Bound Math/Science summer programme for college-intending low-income high school students. The article explains each computer science term with real-life and mathematical examples, applies each term to the Tower of Hanoi puzzle solution, and describes how students connected the terms to their own solutions of the puzzle. It is timely and important to expose mathematics students to computer science concepts. Given the rate at which technology is currently advancing, and our increased dependence on technology in our daily lives, it has become more important than ever for children to be exposed to computer science. Yet, despite the importance of exposing today's children to computer science, many children are not given adequate opportunity to learn computer science in schools. In the United States, for example, most students finish high school without ever taking a computing course. Mathematics lessons, such as the one described in this article, can help to make computer science more accessible to students who may have otherwise had little opportunity to be introduced to these increasingly important concepts.     

The Enantioselective Total Synthesis of Exochomine

Molecules that possess fully substituted chiral centers are often challenging to construct, particularly if those centers connect two seemingly different halves or include a nitrogen atom. Herein, we describe an efficient approach to a molecule that combines both challenges in a single center in the form of exochomine. Failures in direct coupling led to a design fueled by highly specific reaction conditions for several steps and the development of an improved protocol for 1,4‐reduction in a hindered context where numerous side reactions were possible. These chemoselective solutions should have value to other problems. Challenges in obtaining matching spectral data for the synthesized natural product are also discussed.     

Polarizing the Nazarov cyclization: the impact of dienone substitution pattern on reactivity and selectivity

The impact of dienone substitution on the Nazarov cyclization has been examined in detail. Substrates bearing different substituents at each of four positions on the dienone backbone were systematically probed in order to identify trends leading to higher reactivity and better selectivity. Desymmetrization of the pentadienyl cation and oxyallyl cation intermediates through placement of polarizing groups at both the C-2 and C-4 positions was found to be particularly effective. These modifications allowed cyclizations to occur in the presence of catalytic amounts of mild Lewis acids. It was also found that stereoconvergent cyclization of mixtures of E and Z isomers of alkylidene beta-ketoesters occurred via an efficient isomerization process that occurred under the reaction conditions.     

Effects of a Tridentate Pincer Ligand on Parahydrogen Induced Polarization

The role of ligands in rhodium- and iridium-catalyzed Parahydrogen Induced Polarization (PHIP) and SABRE (signal amplification by reversible exchange) chemistry has been studied in the benchmark systems, [Rh(diene)(diphos)]⁺ and [Ir(NHC)(sub)₃(H)₂]⁺, and shown to have a great impact on the degree of hyperpolarization observed. Here, we examine the role of the flanking moieties in the electron-rich monoanionic bis(carbene) aryl pincer ligand, ^ArCCC (Ar=Dipp, 2,6-diisopropyl or Mes, 2,4,6-trimethylphenyl) on the cobalt-catalyzed PHIP and PHIP-IE (PHIP via Insertion and Elimination) chemistry that we have previously reported. The mesityl groups were exchanged for diisopropylphenyl groups to generate the (^DippCCC)Co(N₂) catalyst, which resulted in faster hydrogenation and up to 390-fold ¹H signal enhancements, larger than that of the (^MesCCC)Co-py (py=pyridine) catalyst. Additionally, the synthesis of the (^DippCCC)Rh(N₂) complex is reported and applied towards the hydrogenation of ethyl acrylate with parahydrogen to generate modest signal enhancements of both ¹H and ¹³C nuclei. Lastly, the generation of two (^MesCCC)Ir complexes is presented and applied towards SABRE and PHIP-IE chemistry to only yield small ¹H signal enhancements of the partially hydrogenated product (PHIP) with no SABRE hyperpolarization.     

Origins of stereoselectivity in the oxido-alkylidenation of alkynes

A mild, convenient oxido-alkylidenation of alkynes is described. The three-step sequence involves the 1,3-dipolar cycloaddition of a nitrone and an alkynoate, oxidation of the resulting isoxazoline, and stereoselective extrusion of nitrosomethane. Quantum mechanical calculations identified the interactions of R3 with the oxidant and the preferred conformation of a diradical intermediate as major factors controlling the stereoselectivity of the oxidation and torquoselectivity of the extrusion.     

Spatial dynamics of steady flames 2. Low-dimensional manifolds and the role of transport processes

The study of the spatial dynamics of steady one-dimensional H 2/O 2 flames is continued. Algorithms for generating low-dimensional manifolds for these systems are presented and used to find low-dimensional manifolds for the flames and the corresponding adiabatic, isobaric chemical-kinetic systems. It is demonstrated that these algorithms generate manifolds that are more accurate than the ILDM algorithm for two-dimensional manifolds of the flames. The manifolds are then employed to study the relationship between the manifolds of the flame and the manifolds of the chemical-kinetic system. It is shown that the one-dimensional manifolds of the flame match well with the composite manifolds of the chemical kinetics, but that for two-dimensional manifolds there are discrepancies between the flame manifolds and the chemical-kinetic manifolds.     

Anion-arene adducts: C-H hydrogen bonding, anion-pi interaction, and carbon bonding motifs

This article summarizes experimental and theoretical evidence for the existence of four distinct binding modes for complexes of anions with charge-neutral arenes. These include C-H hydrogen bonding and three motifs involving the arene-pi system-the noncovalent anion-pi interaction, weakly covalent sigma interaction, and strongly covalent sigma interaction.     

Exploration of variables in the fabrication of pyrolysed photoresist

The increased use of pyrolysed photoresist films (PPF) as electrode material intensifies the need to know what fabrication variables are important in the pyrolysis process. The main factor effects of seven variables (three time/temperature heating levels plus the position in the furnace tube) in the fabrication of PPF were investigated by a Plackett and Burman, eight-run, experimental design. In the three-step pyrolysis programme, gas flow had a large effect on the surface cleanliness and roughness. It was also observed that the position in the furnace affected the resistivity of the PPF. Fabrication parameters that give rapid electron transfer to redox species in solution, that provide low surface oxide and that lead to low surface roughness were identified. The guidelines on what fabrication conditions to employ to give a variety of different electrode characteristics are presented.