Cellular Automata, the Collatz Conjecture and Powers of 3/2

  • Jarkko Kari
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)


We discuss one-dimensional reversible cellular automata F ×3 and F ×3/2 that multiply numbers by 3 and 3/2, respectively, in base 6. They have the property that the orbits of all non-uniform 0-finite configurations contain as factors all finite words over the state alphabet {0,1,…,5}. Multiplication by 3/2 is conjectured to even have an orbit of 0-finite configurations that is dense in the usual product topology. An open problem by K. Mahler about Z-numbers has a natural interpretation in terms the automaton F ×3/2. We also remark that the automaton F ×3 that multiplies by 3 can be slightly modified to simulate the Collatz function. We state several open problems concerning pattern generation by cellular automata.


Cellular Automaton Product Topology Tiling Problem Universal Pattern Forward Orbit 
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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jarkko Kari
    • 1
  1. 1.Department of MathematicsUniversity of TurkuTurkuFinland

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