simple magic). Several of them may be pandiagonal magic. Gardner also called this (Langman’s pandiagonal) a 'perfect' cube, presumably not realizing it was a higher class then Myer’s cube. See previous note re Boyer and Trump. The smallest normal pandiagonal magic cube is order 7; see Pandiagonal magic cube.Minimum correct summations required = 9m2 + 4. All pan-r-agonals sum correctly for r = 1 and 2. Perfect: All 3m planar arrays must be pandiagonal magic squares. In addition, all pantriagonals must sum correctly. These two conditions combine to provide a total of 9m pandiagonal magic squares. The smallest normal perfect magic cube is order 8; see Perfect magic cube. Nasik; A. H. Frost (1866) referred to all but the simple magic cube as Nasik! C. Planck (1905) redefined Nasik to mean magic hypercubes of any order or dimension in which all possible lines summed correctly. i.e. Nasik is a preferred alternate, and less ambiguous term for the perfect class.Minimum correct summations required = 13m2. All pan-r-agonals sum correctly for r = 1, 2 and 3. Alternate Perfect Note that the above is a relatively new definition of perfect. Until about 1995 there was much confusion about what constituted a perfect magic cube (see the discussion under Diagonal). Included below are references and links to discussions of the old definition With the popularity of personal computers it became easier to examine the finer details of magic cubes. Also more and more work was being done with higher-dimension magic hypercubes. For example, John Hendricks constructed the world's first Nasik magic tesseract in 2000. Classed as a perfect magic tesseract by Hendricks definition. Generalized for All Dimensions A magic hypercube of dimension n is perfect if all pan-n-agonals sum correctly. Then all lower-dimension hypercubes contained in it are also perfect. For dimension 2, The Pandiagonal Magic Square has been called perfect for many years. This is consistent with the perfect (Nasik) definitions given above for the cube. In this dimension, there is no ambiguity because there are only two classes of magic square, simple and perfect. In the case of 4 dimensions, the magic tesseract, Mitsutoshi Nakamura has determined that there are 18 classes. He has determined their characteristics and constructed examples of each. And in this dimension also, the Perfect (Nasik) magic tesseract has all possible lines summing correctly and all cubes and squares contained in it are also Nasik magic. Another