  There are 4 x 9 = 36 entries in this table. These entries are taken from a set of 18 rays and every ray appears twice.  It is easy to verify that every column in the table represents a set of four orthogonal rays.  Since there are 9 columns, we must end up with an odd number of the table’s entries coloured white. However, since every ray appears twice any time we colour one of them white, we commit ourselves to colouring an even number of the entries white. It follows that the total number of table entries coloured white must be even, not odd. Thus, a colouring of these 18 rays in accordance with VC1′ is impossible.  (Note for future reference that the first part of the argument — the argument for ‘odd’ — uses only VC1′, while the second — the argument for ‘even’ — relies essentially on NC, by assuming that occurrences of the same ray in different columns are assigned the same number!)