A magic square is an n × n array of distinct positive integers whose sum along any row, column, or main diagonal is the same number. We compute the number of such squares for n = 4, as a function of either the magic sum or an upper bound on the entries. The previous record for both functions was the n = 3 case. Our methods are based on inside-out polytopes, i.e., the combination of hyperplane a...

The cooperative acquisition, defense and distribution of meat are common problems that must be solved by social hunters. These were also problems faced by any of our hominid ancestors that may have hunted large game in our evolutionary past. Much of the last 40 years of research in evolutionary ecology has tried to understand such cooperative behavior. The classic Prisoner's Dilemma game theory...

background: accurate small radiation field dosimetry is essential in modern radiotherapy techniques such as stereotactic radiosurgery (srs) and intensity modulated radiotherapy (imrt). precise measurement of dosimetric parameters such as beam profile, percentage depth doses and output factor of these beams are complicated due to the electron disequilibrium and the steep dose gradients. in the p...

A magic square of order n consists of the numbers 1 to n placed such that the sum of each row, column and principal diagonal equals the magic sum n(n +1)/2. In addition, an odd ordered magic square is associative or self-complementary if diagonally opposite elements have the same sum (n +1)/2. The magic square is said to be regular Greco-Latin if it can be decomposed as a sum of a pair of Latin...

Physics interpretation of the " magic " baseline, L magic , that can play important role in future oscillation experiments is given. The " magic " baseline coincides with the refraction length, l 0. The latter, in turn, approximately equals the oscillation length in matter at high energies. Therefore at the baseline L = l 0 the oscillation phase is 2π, and consequently, the " solar " amplitude ...