Rulesets for Beatty games

We describe a ruleset for a 2-pile subtraction game with P-positions \(\{(\lfloor \alpha n \rfloor ,\lfloor \beta n \rfloor ) : n \in \mathbb Z_{\ge 0} \}\) for any irrational \(1< \alpha < 2\), and \(\beta \) such that \(1/\alpha +1/\beta = 1\). We determine the \(\alpha \)’s for which the game can be represented as a finite modification of t-Wythoff (Holladay, Math Mag 41:7–13, 1968; Fraenkel, Am Math Mon 89(6):353–361, 1982) and describe this modification.     

Structural distortions and dynamic behavior of the elusive "L"-shaped cis-bicyclo[3.3.0]octenyllithium: X-ray crystallographic and NMR studies

Substituted cis-bicyclo[3.3.0]octenyllithium prepared by addition of t-BuLi to 3-methylene-1,4-cyclooctadiene in the presence of TMEDA crystallizes as a dimer with one unsolvated Li(+) sandwiched between the external faces of two allyl anions in a triple ion, and external to it the second Li(+) is bidentately complexed to TMEDA, 8. Within each allyl unit, the allyl bonds have different lengths, and all four rings deviate from coplanarity which relieves strain in the rings despite introducing partial localization of the allyl anions. A similar structure prevails in solution as shown by (7)Li NMR and the results of (7)Li{(1)H} HOESY and (1)H, (1)H NOESY experiments. Carbon-13 NMR line shape changes indicate that the system undergoes a fast allyl bond shift concerted with conformation shifts of the out of plane carbons, ca. DeltaG = 9 kcal x mol(-1). Cyclopentyllithium prepared by CH(3)Li cleavage of the trimethylstannyl derivative slowly undergoes an allowed ring opening to pentadienyllithium as well as deprotonating the solvent. The different behavior of dienylic lithium species is attributed to the relative separation of their termini.     

A One‐Pot Organometallic Route from a 4‐Alkenylpyridine to a 4,4‐Spiro‐Linked Ethyl 1,4‐Dihydropyridine‐1‐carboxylate

In a one‐pot process without isolation of intermediates, (but‐3‐en‐1‐yl)pyridine ( 13 ) is treated sequentially with dicyclohexylborane, trimethylaluminium, and ethyl carbonochloridate yielding ethyl 1,4‐dihydro‐4,4‐(tetramethylene)pyridine‐1‐carboxylate (=ethyl 8‐azaspiro[4.5]deca‐6,9‐diene‐8‐carboxylate; 2 ) in 46% yield based on starting alkenylpyridine 13 (Scheme 5).     

The number of fixed points of the majority rule

Formulae are derived for the number of cyclic binary strings of length n in which no single 1 occurs between two zeros and no single 0 occurs between two ones, and for the number of cyclic binary strings without substrings of the form 000 and 111. This problem is motivated by a problem of genetic information processing.