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Long one-dimensional magic-integer sequences are used to express the phases of 10-20 primary reflexions. The magic-integer representation of phases is extended to other secondary reflexions through strong triplephase relationships involving one secondary and two primary reflexions. In the MAGEX procedure multiple magic-integer representations of the secondaries are sought and the error involved in their subsequent use in a conventional ~ map is much reduced. In view of the large number of primary reflexions the indices of the terms included in the ~' map are large and maps may be computed at up to 220 points. Further reflexions, in batches of ten or so, may be added to the initial set by the further use of magic integers and small-scale maps. When the base of estimated phases is sufficiently large then the phase information is extended by the controlled use of the tangent formula. Examples of the successful application of MA GEX are described.

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