Sorry, it appears my "proof sketch" of the impossibility theorem 3 was too facile... though probably the same general idea just done more carefully, will prove it... I hope. The other theorems there still are ok, far as I currently know... By using mixed-radix number systems instead of base 10, the einstein-tile construction can be redone in any of a continuum-infinite number of genuinely different ways. (Can this infinity be boosted to an even higher infinity?) It also is interesting to note that the tilings suggested are not even a "quasicrystal." As a sanity check (always a good idea...) it might be interesting to draw a picture of one of them. I guess the nicest (?) is the version based on radix=4 at all positions. In the picture, if we color touching-tiles differently, then how many colors will be required? For each K>0 there is a tile which touches at least K others (as you can see by considering the numbers ending in a <=K-long string of 9s or string of 0s...). But this alone does not suffice to show we need infinity colors. And in fact I think every tile only touches a finite number of others with the "average" number of neighbors being bounded, which suggests that only a small finite number of colors is required.