Illustrated here is Nancy Cabot's version of TILE PUZZLE, and which debuted in the Chicago Tribune on March 1, 1935. There is a Ladies Art Company Catalogue version (#289, titled PUZZLE TILE, published 1897), with double repeats, rather than this more elegant version beautifully simplified. To get even simpler, Barbara Brackman has a gorgeous section devoted to designs where one block is a checkerboard, (see her ENCYCLOPEDIA OF PIECED QUILT PATTERNS, pp.141-2).
Cabot's notes state the following:
"[TILE PUZZLE] is another example of what may be done with two simple patches — a square and a triangle. The result is effective; it is easily pieced and works up quickly. The original quilt from which this pattern was taken was made as an all-over pieced quilt, and it was indeed a puzzle to determine the individual blocks in the ensemble. A modern coverlet may be made of alternate pieced and plain blocks and be equally beautiful and, perhaps, not quite so puzzling."
More patterns at this site especially characterized by the elegance of their simplicity would include among others:
HOMESPUN BLOCK and
STAR AND BOX, and maybe the most perfect,
A THOUSAND PYRAMIDS.
For comparative designs where one block is a checkerboard, see:
AUNT PATSY'S PET