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By Martin Gardner

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The matter contains 8 women and 8 days. 148 bankruptcy 11 1 2 three ladies four five 6 7 eight determine sixty five A Hadamard chessboard quantity the rows and columns as proven. The 8th lady is an older lady who chaperones the triplets day-by-day, and at the 8th day all 8 women cross into city. for every day (indicated via a row), colour the cells that point out the 3 ladies (plus the chaperone) who stroll to city. the end result: a Hadamard matrix! there's a uncomplicated process for producing Hadamard matrixes for all orders which are powers of 2 (see determine 66). The order-2 development is positioned in 3 corners of the order-4 sq., and its unfavorable (colors reversed) is going into the decrease correct nook. a similar strategy with the order-4 trend generates the order eight, and so forth for the better powers. extra often, given Hadamard matrixes of orders m and n, a matrix of order mn should be created just by changing every one coloured mobilephone of m via the total development of n and every uncolored mobile of m by means of the adverse of n. the massive matrix is termed the tensor made of the 2 smaller ones. It doesn't topic even if m is greater, smaller, or equivalent to n. If, for instance, you might have a Hadamard matrix of order 12, the tensor made from such matrixes is a Hadamard matrix of order a hundred and forty four. Hadamard matrixes are greater than playthings. they're used for the development of precious error-correcting binary codes. For descriptions of such functions, the reader is noted the hot version of Ball's publication (so seriously revised as to be nearly a brand new paintings) and to the references it cites. whilst the Mariner spacecraft of 1969 despatched again images of Mars, Coxeter tells us, they have been despatched in an error-correcting code in keeping with the order-8 Hadamard matrix. In remaining, allow us to go back to the Csaszar polyhedron and pose an exciting challenge. The Csaszar toroid can't be built if all its faces are equilateral triangles. consider a one-hole toroid is made solely of congruent trian- THE CSASZAR POLYHEDRON 149 gular faces, all equilateral. what's the minimal variety of faces it may have? Bonnie M. Stewart, a mathematician at Michigan country collage, considers this challenge on web page forty eight of his booklet Adventures one of the Toroids (1970). Stewart offers the development of this type of torus with fifty-four faces, twentyseven vertexes, and eighty-one edges. one in every of his scholars, Kurt Schmucker, chanced on a one-hole toroid with forty-eight equilateral triangular faces. no matter if that's the minimal, besides the fact that, is one other tantalizing, unanswered toroidal query. solutions the original answer for the Steiner triple process of order nine is received from the subsequent matrix: a b c d e f g h i eight determine sixty six Hadamard matrixes for powers of 2 150 bankruptcy 11 The rows (abc, def; g h i ) offer 3 triplets. The columns (adg, beh, cf;) offer 3 extra. the most diagonals (aei, ceg) and the damaged diagonals (bfg, cdh, afh, bdi) whole the checklist of twelve triplets. The orders of Steiner triple platforms are numbers that experience a rest of 1 or three once they are divided by means of 6.

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