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Examples of computations being done by Turing machines with two states and two colors. Evolution from a succession of initial conditions is shown corresponding to inputs of numbers from 1 to 20. Each block of Turing machines yields the same output for a given input. A computation is taken to be complete when the head of the Turing machine goes further to the right than it was at the beginning. The plots show how many steps this takes for successive inputs with lengths up to 9. The maximum for input of length n is (a) 5, (b) 6n+3, (c) 4n+3, (d) 2n+3, (e) 2n^2 + 8n + 7, (f) 2^(n+1)-1 (though the average is n+2), (g) 2n+1, (h) 3, (i) 2n+1, (j) 4n-1, (k) roughly 2.5 n^2. In cases (i), (j) and (k) there are some inputs for which the head goes further and further to the left, and the Turing machine never halts. The machines shown are numbered 3279, 1285, 3333, 261, 1447, 1953, 1969, 3517, 3246, 3374, 1507.

From Stephen Wolfram: A New Kind of Science [citation]