Puzzle 25:  Rational Creatures

The DNA of a randimal is an infinite string of 0’s and 1’s and is formed as follows: The randoplast produces a random string of 100 bits. Then the string doubles itself in length at times 2−k for k = 1, 2, . . . , seconds. The whole sequence is completed after one second This doubling in length goes as follows:
If the string is currently x1, x2, . . . , xm then the doubled string is x1, x2, · · · , xm, 1 − x1, 1 − x2, · · · 1 − xm.
The behaviour of the creature is rational or irrational, depending on whether the DNA defines a rational or irrational number. Are there any rational randimals?




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