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YAP: Yet Another Puzzle

YAP: Yet Another Puzzle. Boudewijn R. Haverkort November 16, 2007. Collatz Conjecture (1937). n = 1, 2, 3,... Sequence of n such that n := n /2 if n is even n := 3 n + 1, if n is odd Conjecture: for all n ≥ 1, the above sequence returns to 1 “Always Eventually 1”.

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YAP: Yet Another Puzzle

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  1. YAP: Yet Another Puzzle Boudewijn R. Haverkort November 16, 2007

  2. Collatz Conjecture (1937) • n = 1, 2, 3,... • Sequence of n such thatn := n/2 if n is evenn := 3n + 1, if n is odd • Conjecture: for all n ≥ 1, the above sequence returns to 1 • “Always Eventually 1”

  3. A few cases... • Sequence of n such thatn := n/2 if n is evenn := 3n + 1, if n is odd • 5, 16, 8, 4, 2, 1 • 6, 3, 10, 5, 16, 8, 4, 2, 1 • 27, 82, 41, ...111 steps..., 4, 2, 1 • Computer-checked up to 2.88 x 1018

  4. Probabilistic interpretation • View sequence of n’s as DTMC on the positive natural numbers, with “probability 1” transitions • pCTL-type of model checking: -- find the satisfaction set for P≥1{AE “1”}-- this set should be equal to { 1, 2,... } • Use all we know about Markov chains, abstraction, bisimulation, infinite-state Markov chains, regular model checking,... • //en.wikipedia.org/wiki/Collatz_conjecture

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