Showing papers by "Hermann Bondi published in 1982"
15 Sep 1982-Philosophical transactions - Royal Society. Mathematical, physical and engineering sciences
TL;DR: A brief history of work on the 4 x 4 magic square is presented in this article, with particular reference to Frenicle's achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 1976).
Abstract: A brief history of work on the 4 x 4 magic square is presented, with particular reference to Frenicle’s achievement over 300 years ago of establishing 880 as the number of essentially different squares by using the method of exhaustion (not convincingly repeated except by computer in 1976) He also established several central theorems Our paper confirms the number 880 by a wholly new method of Frenicle quads and ‘part sums’, which leads to the classification of all solutions into, initially, six genera one of which has no members and thence to the enumeration of all possible solutions by analytical methods only The working leads also to the first analytical proof independent of solutions that 12 and only 12 patterns formed by linking‘complementary’ numbers within a square are necessary and sufficient to describe all solutions - a fact which has been known since 1908, but not hitherto proved A second method of construction and partial proof) greatly shortened by what has gone before, is also described This yields a highly symmetrical list of the 880 magic squares Together the two methods combine to explain many of the special characteristics and otherwise mysterious properties of these fascinating squares The complete symmetrical list of squares ends the paper
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TL;DR: In this article, the sources and costs of scientific knowledge are discussed and two problem areas in the British Government system pointed out; as remedy a 'New Science Education' is advocated.
Abstract: That Government works at all, should surprise us; it invariably has to deal with most difficult questions involving many imponderables, particularly in the technological field. Rendering scientific advice in the Ministry of Defence and other British Government Departments involves the conveying of various shades of uncertainty; invariably a difficult task not made easier by the lack of scientific background of senior politicians and senior civil servants. The sources and the costs of scientific knowledge are here discussed and two problem areas in the British Government system pointed out; as remedy a 'New Science Education' is advocated.
1 citations