We present a new aperiodic tileset containing 11 Wang tiles on 4 colors, and we show that this tileset is minimal, in the sense that no Wang set with either fewer than 11 tiles or fewer than 4 colors is aperiodic. This gives a definitive answer to the problem raised by Wang in 1961.

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An aperiodic set of 11 Wang tiles

Publisher Summary This chapter presents a simplified proof for the insolvability of the decision problem in the case ΛVΛ. Using the ideas of Wang and Buchi, the first proof for the insolvability of the decision problem in the case ΛVΛ has been given by Kahr and others. Wang, who invented the domino games, proved that the decision problem for the corner game is unsolvable. Buchi inferred that the decision problem for the case V Λ ΛVΛ is insolvable. The insolvability of the decision problem for the diagonal game leads to the result that the decision problem for ΛVΛ is unsolvable. A minor change consists in replacing the diagonals by the rows of the first quadrant.

H. Hermes

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A Simplified Proof for the Unsolvability of the Decision Problem in the Case